M  !<  i  DI  ••  i  :.  ,-•;<;   •   ,i    .  , 


IN  MEMORIAM 
FLORIAN  CAJOR1 


SOLID  GEOMETRY 


BY 


HERBERT  E.  HAWKES,  PH.D. 

PROFESSOR  OF   MATHEMATICS   IN  COLUMBIA    UNIVERSITY 


WILLIAM  A.  LUBY,  M.A. 

HEAD  OF  THE   DEPARTMENT   OF   MATHEMATICS   IN   THE 
JUNIOR  COLLEGE   OF    KANSAS   CITY 

AND 

FRANK  C.  TOUTON,  PH.D. 

LECTURER  IN   EDUCATION,  DEPARTMENT  OF   EDUCATION 
UNIVERSITY  OF  CALIFORNIA 


GINN  AND  COMPANY 

BOSTON     •     NEW   YORK     •     CHICAGO     •     LONDON 
ATLANTA     •     DALLAS     •     COLUMBUS     •     SAN   FRANCISCO 


COPYRIGHT,  1922,  BY 

HERBERT  E.  HAWKES,  WILLIAM  A.  LUBY 
AND  FRANK  C.  TOUTON 

ALL   RIGHTS   RESERVED 
322.4 


AJORI 


GINN   AND  COMPANY  •  PRO- 
PRIETORS •  BOSTON  •  U.S.A. 


PREFACE 

During  recent  years  the  study  of  solid  geometry  has  occu- 
pied a  somewhat  less  commanding  position  in  the  mathematical 
curriculum  than  formerly.  Important  and  essential  as  its  sub- 
ject matter  is  admitted  to  be,  it  has  been  little  more  than 
an  appendage  to  plane  geometry,  both  in  the  methods  of  its 
presentation  and  in  its  scientific  results. 

The  authors  of  this  text  feel  that  the  subject  is  much  more 
vital  than  such  a  tendency  would  indicate.  Not  only  are  the 
bare  truths  gained  from  a  study  of  solid  geometry  essential 
to  the  student  of  science,  but  through  its  medium  a  multi- 
tude of  mathematical  ideas  can  be  presented  and  elucidated 
in  a  natural  and  convincing  manner.  In  fact,  no  subject  of 
elementary  mathematics  can  be  compared  to  solid  geometry 
as  a  climax  and  capstone  of  mathematical  study  for  the  stu- 
dent who  pursues  the  subject  no  farther.  It  not  only  utilizes 
and  applies  much  that  he  has  learned  in  other  courses,  but 
serves  as  a  point  of  vantage  from  which  may  be  gained  many 
glimpses  of  scientific  fields  which  he  is  not  to  enter. 

In  this  text  the  authors  have  presented  the  subject  in  such 
form  that  a  minimal  course  as  prescribed  by  the  colleges  and 
the  various  examining  boards  may  be  covered.  At  the  same 
time  it  affords  at  every  turn  a  richness  of  suggestion  and 
development  for  those  who  have  the  time  and  the  inclination 
to  do  more  than  that  minimum. 

One  of  the  important  opportunities  afforded  by  the  study  of 
solid  geometry  is  that  of  using  and  developing  the  scientific 

iii 

91833 1 


iv  SOLID  GEOMETRY 

imagination.  This  text,  through  its  hundreds  of  queries,  aims 
to  encourage  the  student  to  regard  the  subject  not  merely 
as  a  logical  sequence  of  theorems  but  as  a  subject  inviting 
reflection  and  the  play  of  speculation.  These  queries  should 
be  used  in  class  as  a  basis  for  discussion  and  will  be  found 
to  render  the  more  formal  work  not  only  more  interesting 
but  more  intelligible.  If  time  does  not  permit  any  attention 
to  the  queries,  they  may  be  omitted  from  the  class  assignments 
without  disturbing  the  continuity  of  the  subject.  Their  use, 
however,  is  strongly  urged  by  the  authors. 

Exercises  illustrating  or  dependent  upon  the  various  theo- 
rems are  scattered  throughout  the  text  and  afford  as  much 
drill  of  this  kind  as  many  teachers  can  profitably  use.  The 
collections  at  the  end  of  each  book  may  be  regarded  as  sup- 
plementary. Great  care  has  been  exercised  to  provide  a  col- 
lection of  originals  that  is  fresh,  interesting,  not  too  difficult, 
but  illustrating  all  parts  of  the  subject. 

The  assumption  of  Cavalieri's  Theorem  as  a  basis  for  the 
theorems  on  measurement  is  the  result  of  many  years  of  class- 
room experience.  The  simplicity  and  power  of  this  procedure 
should  commend  it  both  to  teachers  and  to  students. 

The  geometry  of  the  sphere  and  its  relation  to  plane  geom- 
etry is  also  elaborated  with  care  and  in  such  a  manner  as  to 
give  the  student  an  insight  into  the  meaning  of  geometrical 
science. 


CONTENTS 

PAGE 

BOOK  VI.    LINES  AND  PLANKS 303 

Parallel  Lines  and  Planes 311 

Perpendicular  Lines  and  Planes    .          324 

Angles  between  Planes 339 

Projections , 349 

Skew  Lines 354 

BOOK  VII.    POLYHEDRONS,  CONES,  AND  CYLINDERS  .     .     .  358 

Volumes 365 

Cylinders 375 

Pyramids 387 

Cones 405 

Polyhedrons       ....,.< 419 

BOOK  VIII.    THE  SPHERE 443 

General  Properties  of  the  Sphere 443 

Measurement  of  the  Sphere 452 

Geometry  on  the  Sphere 464 

INDEX                                                                       493 


REFERENCES  FROM  PLANE  GEOMETRY 
POSTULATES  AND  AXIOMS 

19.  Postulate  I.    There  is  only  one  straight  line  through  two 
points. 

20.  Postulate  II.   Any  geometric  figure  may  be  moved  from 
one  place  to  another  without  changing  its  size  or  shape. 

31.  Axiom  I.    If  equals  are  added  to  equals,  the  results  are 
equal. 

32.  Axiom  II.    (1)  Two  numbers  or  magnitudes  each  equal  to 
a  third  are  equal  to  each  other.    (2)  Two  figures  congruent  to  a 
third  are  congruent  to  each  other. 

36.  Postulate  III.    All  straight  angles  are  equal. 

39.  Axiom  III.  If  equals  are  divided  by  the  same  number,  the 
results  are  equal. 

41.  Postulate  IV.  At  a  given  point  of  a  line,  one  and  only  one 
perpendicular  can  be  drawn  to  the  line. 

45.  Postulate  V.  The  postulate  of  parallels.  Through  a  given 
point  outside  a  line,  one  line  parallel  to  it  exists,  and  only  one. 

51.  Axiom  IV.  If  equals  are  subtracted  from  equals,  the  results 
are  equal. 

65.  Axiom  V.  A  number  may  be  substituted  for  its  equal  in 
any  operation  on  numbers. 

124.  Axiom  VI.  If  equals  are  multiplied  by  equals,  the  results 
are  equal. 

136.  Axiom  VII.    The  whole  is  greater  than  any  of  its  parts. 

137.  Axiom  VIII.    If  the  first  of  three  magnitudes  is  greater 
than  the  second  and  the  second  is  greater  than  the  th'ird,  the 
first  is  greater  than  the  third. 


viii  SOLID  GEOMETRY 

139.  Axiom  IX.    If  the  same  number,  positive  or  negative,  is 
added  to  or  subtracted  from  each  member  of  an  inequality,  the 
results  are  unequal  in  the  same  order. 

140.  Axiom  X.    If  both  members  of  an  inequality  are  multi- 
plied or  divided  by  the  same  positive  number,  the  results  are 
unequal  in  the  same  order. 

141.  Axiom  XI.    If  the  corresponding  members  of  two  or  more 
inequalities  which  are  in  the  same  order  are  added,  the  sums  are 
unequal  in  the  same  order. 

142.  Axiom  XII.    If  unequals  are  subtracted  from  equals,  the 
results  are  unequal  in  the  reverse  order. 

146.  Postulate  VI.    Any  side  of  a  triangle  is  less  than  the  sum 
of  the  other  two  sides. 

DEFINITIONS 

15.  Angle.    A  plane  angle  (symbol  Z.)  is  the  figure  formed  by 
two  rays  which  meet. 

16.  Triangle.    A  triangle  (symbol  A)  is  a  portion  of  a  plane 
bounded  by  three  straight  lines. 

24.  Congruence.  Two  geometric  magnitudes  are  congruent  if 
their  boundaries  can  be  made  to  coincide. 

30.  Isosceles  triangle.  An  isosceles  triangle  is  a  triangle  which 
has  two  equal  sides. 

37.  Perpendicular.  If  one  straight  line  cuts  another  so  as  to 
make  any  two  adjacent  angles  equal,  each  line  is  perpendicular 
(symbol  _L)  to  the  other. 

43.  Parallel  lines.  Parallel  lines  are  lines  that  lie  in  the  same 
plane  and  do  not  meet  however  far  they  are  produced. 

49.  Hypotenuse.  The  hypotenuse  of  a  right  triangle  is  the  side 
opposite  the  right  angle. 

52.  Vertical  angles.  Two  angles  are  vertical  angles  if  the  sides 
of  one  are  the  prolongations  of  the  sides  of  the  other. 


REFERENCES  FROM  PLANE  GEOMETRY     ix 

54.  Transversal.  A  transversal  is  a  line  that  crosses  (cuts  or 
intersects)  two  or  more  lines. 

61.  Supplementary  angles.  One  angle  is  the  supplement  of 
another  if  their  sum  equals  two  right  angles  (or  180°). 

79.  Regular  polygon.    A  regular  polygon  is  a  polygon  all  of 
whose  angles  are  equal  and  all  of  whose  sides  are  equal. 

80.  Diagonal.    A  diagonal  of  a  polygon  is  a  line  joining  any 
two  nonconsecutive  vertices. 

83.  Parallelogram.    A  parallelogram  is  a  quadrilateral  whose 
opposite  sides  are  parallel. 

84.  Rectangle.  A  rectangle  is  a  parallelogram  whose  angles  are 
right  angles. 

107.  Trapezoid.  A  trapezoid  is  a  quadrilateral  two  and  only 
two  of  whose  sides  are  parallel. 

119.  Concurrent  lines.  Three  or  more  lines  which  have  one 
point  in  common  are  said  to  be  concurrent. 

154.  Circle.  A  circle  is  a  closed  plane  curve  every  point  of 
which  is  equally  distant  from  a  point  in  the  plane  of  the  curve. 

187.  Tangent.  A  tangent  to  a  circle  is  a  straight  line  which, 
however  far  it  may  be  produced,  has  only  one  point  in  common 
with  the  circle. 

233.  Altitude  of  a  triangle.  An  altitude  of  a  triangle  is  a  per- 
pendicular from  any  vertex  to  the  side  opposite,  produced  if 
necessary. 

242.  Locus.  A  locus  is  a  figure  containing  all  the  points,  and 
only  those  points,  which  fulfill  a  given  requirement. 

267.  Similar  polygons.  Two  polygons  are  similar  (symbol  ^)  if 
the  angles  of  one  are  equal  respectively  to  the  angles  of  the  other 
and  the  sides  are  proportional  each  to  each. 

310.  Area.  The  area  of  a  plane  figure  is  the  number  which 
expresses  the  ratio  between  its  surface  and  the  surface  of  the 
unit  square. 


x  SOLID  GEOMETRY 

354.  Center  of  polygon.  The  center  of  a  regular  polygon  is  the 
common  center  of  its  inscribed  and  circumscribed  circles. 

363.  Definition  of  IT.  The  number  TT  (pronounced  pi),  used  in 
calculations  on  the  circle,  is  the  number  obtained  by  dividing  the 

0 

circumference  of  a  circle  by  its  diameter ;  that  is,  TT  =  —  •    From 

the  above,  C  =  TrD  or  C  =  2  irR. 

PROPOSITIONS 

25.  If  two  sides  and  the  included  angle  of  one  triangle  are 
equal  respectively  to  two  sides  and  the  included  angle  of  another, 
the  two  triangles  are  congruent. 

27.  Corresponding  parts  of  congruent  figures  are  equal. 

29.  If  two  sides  of  a  triangle  are  equal,  the  angles  "opposite 
them  are  equal. 

33.  If  the  three  sides  of  one  triangle  are  equal  respectively 
to  the' three  sides  of  another,  the  triangles  are  congruent. 

42.  There  is  only  one  perpendicular  from  a  point  to  a  line. 
44.  Two  lines  perpendicular  to  the  same  line  are  parallel. 

46.  If  a  line  intersects  one  of  two  parallel  lines,  it  intersects 
the  other  also. 

47.  If  a  line  is  perpendicular  to  one  of  two  parallel  lines,  it  is 
perpendicular  to  the  other  also. 

50.  Two  right  triangles  are  congruent  if  the  hypotenuse  and 
an. adjacent  angle  of  one  are  equal  respectively  to  the  hypotenuse 
and  an  adjacent  angle  of  the  other. 

53.  If  two  straight  lines  intersect,  the  vertical  angles  are  equal. 

57.  If  two  parallel  lines  are  cut  by  a  transversal,  the  alternate- 
interior  angles  are  equal. 

66.  The  sum  of  the  angles  of  any  triangle  is  two  right  angles. 
76.  If  two  angles  have  their  sides  perpendicular  each  to  each, 
they  are  equal  or  supplementary. 


REFERENCES  FROM  PLANE  GEOMETRY     xi 

82.  If  a  side  and  the  two  adjacent  angles  of  one  triangle  are 
equal  respectively  to  a  side  and  the  two  adjacent  angles  of 
another,  the  triangles  are  congruent. 

85.  The  opposite  sides  of  a  parallelogram  are  equal. 

88.  If  two  sides  of  a  quadrilateral  are  equal  and  parallel,  the 
figure  is  a  parallelogram. 

90.  The  diagonals  of  a  parallelogram  bisect  each  other. 

97.  Two  right  triangles  are  congruent  if  the  hypotenuse  and 
another  side  of  the  first  are  equal  respectively  to  the  hypotenuse 
and  another  side  of  the  second. 

98.  Two  right  triangles  are  congruent  if  a  side  about  the 
right  angle  and  an  acute  angle  of  the  first  are  equal  respectively 
to  a  side  about  the  right  angle  and  a  corresponding  angle  of 
the  second. 

116.  If  a  point  is  on  the  mid-perpendicular  of  a  line,  it  is  equi- 
distant from  the  ends  of  the  line. 

117.  If  a  point  is  equidistant  from  the  ends  of  a  line,  it  is  on 
the  mid-perpendicular  of  the  line. 

118.  Two  points  each  equally  distant  from  the  extremities  of 
a  line  determine  the  mid-perpendicular  of  the  line. 

121.  If  a  point  is  on  the  bisector  of  an  angle,  it  is  equally 
distant  from  the  sides  of  the  angle. 

122.  If  a  point  is  equally  distant  from  the  sides  of  an  angle, 
it  is  on  the  bisector  of  the  angle. 

125.  If  n  is  the  number  of  sides  of  a  convex  polygon  and  s  is 
the  sum  of  its  interior  angles,  then  s  =  (2  n  —  4)  right  angles. 

128.  If  two  lines  are  parallel  to  a  third  line,  the  two  lines  are 
parallel  to  each  other. 

130.  If  a  line  bisects  one  side  of  a  triangle  and  is  parallel  to 
another  side,  it  bisects  the  third  side. 


xii  SOLID  GEOMETRY 

131.  The  line  which  joins  the  mid-points  of  two  sides  of  a  tri- 
angle is  parallel  to  the  third  side  and  equal  to  one  half  of  it. 

133.  The  line  joining  the  mid-points  of  the  nonparallel  sides 
of  a  trapezoid  is  parallel  to  the  bases  and  equal  to  half  their  sum. 

143.  The  medians  of  a  triangle  are  concurrent  in  the  basal 
point  of  trisection  of  each. 

145.  If  two  sides  of  a  triangle  are  unequal,  the  angles  opposite 
them  are  unequal  and  the  greater  angle  lies  opposite  the  greater 
side. 

149.  The  perpendicular  from  a  point  outside  a  straight  line  is 
the  shortest  line  from  the  point  to  the  line. 

151.  If  two  triangles  have  two  sides  of  one  equal  respectively 
to  two  sides  of  the  other  and  the  third  side  of  the  first  greater 
than  the  third  side  of  the  second,  the  included  angle  of  the  first 
is  greater  than  the  included  angle  of  the  second. 

171.  In  the  same  circle  or  in  equal  circles  if  two  central  angles 
are  equal,  their  intercepted  arcs  are  equal. 

174.  In  the  same  circle  or  in  equal  circles,  if  two  arcs  are 
equal,  they  subtend  equal  central  angles. 

178.  In  the  same  circle  or  in  equal  circles,  if  two  arcs  are 
equal,  their  chords  are  equal. 

179.  In  the  same  circle  or  in  equal  circles,  if  two  chords  are 
equal,  their  subtended  arcs  are  equal. 

182.  If  a  line  passes  through  the   center  of  a  circle  and  is 
perpendicular   to   a   chord,  it   bisects   the   chord   and   the  arcs 
subtended  by  it. 

183.  In  the  same  circle  or  in  equal  circles,  if  two  chords  are 
equal,  they  are  equally  distant  from  the  center. 

191.  If  a  line  is  perpendicular  to  a  radius  at  its  outer  extremity, 
it  is  tangent  to  the  circle. 


REFERENCES  FROM  PLANE  GEOMETRY         xiii 

192.  If  a  line  is  tangent  to  a  circle,  it  is  perpendicular  to  the 
radius  drawn  to  the  point  of  tangency. 

193.  If  two  tangents  are  drawn  to  a  circle  from  an  outside  point, 
(1)  the  tangents  are  equal ;  (2)  the  line  joining  the  outside  point 
to  the  center  bisects  the  angle  between  the  tangents  and  the  angle 
between  the  radii  drawn  to  the  points  of  contact. 

201.  If  two  circles  intersect,  the  line  of  centers  bisects  their 
common  chord  at  right  angles. 

202.  If  two  circles  are  tangent  externally  or  internally,  the 
centers  and  the  point  of  tangency  are  in  a  straight  line. 

211.  A  central  angle  is  measured  by  its  intercepted  arc. 

216.  An  inscribed  angle  is  measured  by  one  half  the  number 
of  degrees  in  its  intercepted  arc. 

217.  An  angle  inscribed  in  a  semicircle  is  a  right  angle. 

224.  If  three  points  are  not  in  the  same  straight  line,  one  circle 
and  only  one  can  pass  through  them. 

260.  A  line  parallel  to  one  side  of  a  triangle  and  cutting  the 
other  two  divides  them  into  four  corresponding  segments  which 
are  proportional. 

263.  If  a  line  parallel  to  one  side  of  a  triangle  cuts  the  other 
two  sides,  the  two  sides  are  in  proportion  to  their  corresponding 
segments. 

265.  If  a  line  divides  two  sides  of  a  triangle  into  proportional 
corresponding  segments,  it  is  parallel  to  the  third  side. 

269.  Corresponding  sides  of  similar  polygons  are  in  proportion. 

270.  If  two  triangles  are  mutually  equiangular,  they  are  similar. 

271.  A  line  cutting  two  sides  of  a  triangle  and  parallel  to  the 
third  side  forms  a  second  triangle  similar  to  the  first. 

277.  In  two  similar  triangles  any  two  homologous  sides  are 
proportional  to  (1)  two  corresponding  altitudes ;  (2)  two  corre- 
sponding medians  ;  (3)  the  bisectors  of  two  corresponding  angles. 


xiv  SOLID  GEOMETRY 

282.  If  a  perpendicular  is  drawn  from  the  vertex  of  the  right 
angle  to  the  hypotenuse  of  a  right  triangle,  (1)  the  two  triangles 
formed  are  similar  to  each  other  and  to  the  given  triangle  ;  (2)  the 
perpendicular  is  a  mean  proportional  between  the  segments  of 
the  hypotenuse ;  and  (3)  the  square  of  either  side  about  the  right 
angle  equals  the  product  of  the  whole  hypotenuse  and  the  seg- 
ment adjacent  to  that  side. 

284.  In  any  right  triangle  the  square  of  the  hypotenuse  equals 
the  sum  of  the  squares  of  the  other  two  sides. 

292.  If  from  a  point  without  a  circle  a  secant  terminating  in 
the  circle  and  a  tangent  be  drawn,  the  square  of  the  tangent  equals 
the  whole  secant  times  its  external  segment. 

312.  The  area  of  a  rectangle  is  the  product  of  its  base  and 
altitude. 

319.  The  area  of  a  parallelogram  is  the  product  of  its  base 
and  altitude. 

320.  The  area  of  a  triangle  is  one  half  the  product  of  its  base 
and  altitude. 

322.  The  area  of  a  trapezoid  is  one  half  the  product  of  its 
altitude  and  the  sum  of  its  bases. 

325.  If  two  triangles  have  an  angle  of  one  equal  to  an  angle 
of  the  other,  their  areas  are  to  each  other  as  the  product  of  the 
sides  including  the  angle  of  the  first  is  to  the  product  of  the  sides 
including  the  angle  of  the  second. 

326.  The  areas  of  two  similar  triangles  are  to  each  other  as 
the  squares  of  any  two  homologous  sides. 

327.  The  areas  of  two  similar  convex  polygons  are  to  each 
other  as  the  squares  of  any  two  homologous  sides. 

359.  The  area  of  a  regular  polygon  is  one  half  the  product  of 
its  perimeter  and  its  apothem. 

369.  The  area  of  the  sector  of  a  circle  equals  one  half  the 
product  of  its  radius  and  its  arc. 


REFERENCES  FROM  PLANE  GEOMETRY         xv 

371.  The  area  of  a  circle  is  TrR2. 


372.  The  areas  of  two  circles  are  to  each  other  as  the  squares 
of  their  radii. 

373.  The  area  of  a  sector  is  to  the  area  of  the  circle  as  the 
angle  of  the  sector  is  to  four  right  angles. 

CONSTRUCTIONS 

232.  Construct  a  triangle,  given  the  three  sides. 

234.  At  a  given  point  in  a  line  construct  a  perpendicular  to 
the  line. 

235.  From  a  given  point  outside  a  line  construct  a  perpen- 
dicular to  the  line. 


SOLID  GEOMETRY 

BOOK  VI 

LINES  AND  PLANES 

377.  Introduction.    Solid  geometry  is  concerned  with  the 
properties  and  relations  of  figures  which  occupy  space.    This 
does  not  imply  that  the  cubes,  cylinders,  and  other  bodies  con- 
sidered in  this  text  are  made  of  wood  or  other  material  sub- 
stance, any  more  than  that  the  squares  and  triangles  of  plane 
geometry  are  made  of  chalk  or  of  the  carbon  from  our  pencils. 
In  both  cases  the  diagrams  or  solids  which  we  construct  are 
merely  rough  aids  to  our  imagination,  helping  us  to  visualize 
the  properties  of  the  real  figures,  which  are  in  every  case 
objects  of  thought  without  material  existence.   Since,  however, 
many  of  the  objects  around  us  very  closely  approximate  the 
form  of  the  purely  geometric  solids,  the  subject  of  solid  geom- 
etry finds  abundant  application  in  the  affairs  of  everyday  life. 

378.  Assumptions.   The  assumptions  of  plane  geometry  fall 
into  two  classes,  the  axioms  and  the  postulates.   Axioms  I-XII 
do  not  refer  either  to  the  plane  or  to  space,  but  to  relations 
between  numbers,  and  will  be  assumed  without  discussion  in 
what  follows.     The  following  assumptions   of  geometric  con- 
tent have  been  made  in  plane  geometry. 

Postulate  I.    There  is  only  one  straight  line  through  two  points 

(§19)- 

Postulate  II.    Any  geometric  figure  may  be  moved  from  one 

place  to  another  without  changing  its  size  or  shape  (§20). 

303 


304     ,  V  SOLID  GEOMETRY 

Postulate  .V>  ,'  T'lirouyJi  a  given  point  outside  a  line  one  line 


parallel  to  it  exists,  and  only  one  (§45). 
These  postulates  are  assumed  to  bold  in  space. 

Postulate  I  may  also  be  restated  as  follows  :  Two  points  deter- 
mine a  straight  line.  One  point  does  not  determine  a  line,  because 
more  than  one  line  passes  through  a  fixed  point.  Nor  do  three 
points  determine  a  line,  because  no  line  can  be  found  which  passes 
through  any  three  points  taken  at  random.  The  significance  of 
the  word  determine  in  this  statement  is  that  there  is  one  and  only 
one  line  which  contains  the  two  points. 

In  several  theorems  of  plane  geometry,  Postulate  II  is  assumed 
to  hold  in  space.  For  example,  in  proving  that  two  triangles  are 
congruent  if  three  sides  of  one  are  equal  respectively  to  three 
sides  of  the  other  (§  33),  the  triangle  may  have  to  be  lifted  out  of 
..the  plane  and  turned  over  in  space,  in  order  to  make  it  take  the 
.desired  position. 

The  word  move  in  Postulate  II  does  not  imply  that  one  can  lift 
the  figures  of  solid  geometry,  as  one  would  lift  stone  blocks,  and 
carry  them  from  place  to  place.  One  cannot  be  expected  to  move 
in  this  sense  anything  that  is  merely  an  object  of  thought.  To 
move  a  figure  in  geometry  is  to  transfer  our  attention  from  a 
figure  in  one  position  to  another  figure  exactly  like  it  somewhere 
else.  Hence  congruent  figures  in  space  as  well  as  in  a  plane  may 
be  considered  as  the  same  figure  in  different  positions. 

379.  Definitions  from  plane  geometry.  The  definitions  given 
in  plane  geometry  will  be  taken  over  to  solid  geometry  with- 
out change.  One  must  observe,  however,  that  in  solid  geom- 
etry the  emphasis  in  certain  definitions  is  entirely  redistributed. 
For  example,  parallel  lines  are  defined  (§  43)  as  lines  that  lie 
in  the  same  plane  and  do  not  meet  however  far  they  are  pro- 
duced. In  the  study  of  plane  geometry  it  is  never  necessary  to 
emphasize  the  clause  "  that  lie  in  the  same  plane,"  because  in 
every  case  the  whole  figure  lies  in  one  plane.  But  in  solid  geom- 
etry, where  our  figures  lie  anywhere  in  space,  in  proving  two 


BOOK  VI  305 

lines  parallel  it  is  frequently  more  difficult  to  prove  that  they 
lie  in  the  same  plane  than  it  is  to  show  that  they  cannot  meet 
however  far  they  are  produced. 

QUERY  1.    Are  two  horizontal  lines  necessarily  parallel?    Illustrate. 
QUERY  2.    Are  two  vertical  lines  necessarily  parallel?    Illustrate. 
QUERY  3.    Is  every  pair  of  lines  in  space  either  intersecting  or  paral- 
lel ?    Illustrate. 

380.  Undefined  terms.    It  is  assumed  that  the  terms  figure, 
line,  curve,  surface,  and  solid  are  familiar. 

The  word  line  will  be  used  to  denote  a  straight  line  extend- 
ing indefinitely  in  both  directions.  To  avoid  ambiguity,  the 
portion  of  a  line  between  two  of  its  points  will  often  be  called 
a  line-segment. 

381.  The  plane.   A.  plane  is  a  surface  such  that  if  any  two 
points  in  it  are  taken,  the  straight  line  containing  them  .lies 
wholly  in  the  surface. 

Since  a  line  extends  indefinitely  in 
both  directions,  a  plane  is  unlimited  in 
extent.  On  account  of  the  size  of  our 
page  it  is  impossible  to  show  the  whole  plane  by  a  drawing. 
Hence  it  is  customary  to  draw  a  parallelogram  to  represent  a 
plane  or  that  part  of  a  plane  with  which  we  are  particularly 
concerned.  If  it  is  desired  to  emphasize  the  fact  that  the 
plane  extends  farther  in  a  certain  direction,  one  side  of  the 
parallelogram  .may  be  replaced  by  a  wavy  line. 

QUERY  4.    What  postulate  is  assumed  in  the  definition  of  §  381. 

QUERY  5.  Show  from  the  definition  that  two  walls  of  a  room  do  not 
form  a  single  plane. 

QUERY  6.  Is  the  surface  of  a  perfectly  calm  sea  a  plane? 

QUERY  7.  Do  two  points  exist  on  the  curved  surface  of  a  straight  flag- 
pole such  that  the  line  passing  through  them  lies  entirely  in  the  surface 
of  the  pole  ?  If  so,  why  is  this  surface  not  a  plane  ? 

QUKRY  8.    Does  a  plane  have  any  edges? 


306  SOLID  GEOMETRY 

382.  Postulate  VII.    Two  intersecting  lines  lie  in  one  and  in 
only  one  plane. 

This  postulate  plays  about  the  same  role  in  solid  geometry  that 
Postulate  I  does  in  plane  geometry. 

QUERY  1.  Can  more  than  one  plane  pass  through  a  given*  line  ? 
Illustrate.  Does  one  line  determine  a  plane  ? 

QUERY  2.  Can  you  hold  two  pencils  in  such  positions  as  to  show 
that  a  plane  cannot  contain  any  two  lines  taken  at  random  in  space? 

QUERY  3.  Does  a  set  of  three  concurrent  lines  necessarily  determine 
a  plane?  Illustrate. 

QUERY  4.   Is  a  triangle  necessarily  a  plane  figure?   Why? 

Theorem  1 

383.  Three  points  not  on  the  same  line  lie  in  one  and  in 
only  oneplane. 


Given  three  points  A,  £,  and  C  which  do  not  lie  on  the  same 
line. 

To  prove  that  A,  B,  and  C  lie  in  one  and  in  only  one  plane. 

Proof.  Draw  the  lines  AB  and  AC. 

Denote  by  M  the  plane  containing  them  both.  §  382 

A ,  B,  and  C  lie  in  the  plane  M,  since  the  lines  containing  them 
lie  in  that  plane. 

A,  B)  and  C  cannot  lie  in  any  other  plane  than  M,  since  the 
lines  containing  them  cannot  lie  in  another  plane.  §  382 

Hence     A,  B,  and  C  lie  in  one  and  in  only  one  plane. 

*  When  a  figure  is  referred  to  as  ff  given,"  it  is  understood  that  the  figure 
is  supposed  to  be  fixed  both  in  size  and  in  position.  Thus,  a  given  circle  is 
one  of  definite  size  which  is  assumed  to  be  fixed  in  position  during  the  dis- 
cussion. Of  course,  a  line  or  a  plane  can  be  fixed  only  in  position,  since  by 
definition  they  are  indefinite  in  extent. 


BOOK  VI  307 

384.  Corollary  1.  A  line  and  a  point  not  on  the  line  lie  in  one  and 
in  only  one  plane. 


Given  the  line  a  and  the  point  P  not  on  a. 

To  prove  that  P  and  a  lie  in  one  and  in  only  one  plane. 

Proof.  Let  K  and  L  be  any  two  points  on  a. 
Denote  by  M  the  only  plane  containing  P,  K,  and  L.  §  383 

The  line  a  must  lie  in  M.  §  381 

Hence      both  a  and  P  lie  in  M  and  in  no  other  plane. 

385.   Corollary  2.    Two  parallel  lines  lie  in  one  and  in  only 
one  plane. 


Given  the  parallel  lines  a  and  b. 

To  prove  that  a  and  b  'lie  in  one  and  in  only  one  plane. 

Proof.  a  and  b  lie  in  one  plane.  §  43 

If  they  could  lie  in  two  planes  at  once,  then  one  of  them  and 
any  point  of  the  other  would  lie  in  two  planes  at  once,  which 
is  impossible.  §  384 

Hence         a  and  b  lie  in  one  and  in  only  one  plane. 

386.  Restatement.  The  foregoing  results  may  be  stated 
as  follows: 

1.  Two  intersecting  lines  determine  a  plane. 

2.  Three  points  not  in  the  same  line  determine  a  plane. 

3.  A  line  and  a  point  outside  the  line  determine  a  plane. 

4.  Tivo  parallel  lines  determine  a  plane. 


308  SOLID  GEOMETRY 

387.  Perspective.  In  the  diagrams  of  solid  geometry  the 
figure  is  usually  supposed  to  be  either  to  the  right  or  the 
left  of  the  eye  of  the  observer  or  directly  in  front  of  and 


slightly  below  it.    When  a  figure  is  drawn  as  it  appears  to 
the  eye,  it  is  said  to  be  drawn  in  perspective. 

QUERY  1.  What  is  the  position  of  the  three  preceding  figures  with 
respect  to  the  eye  of  the  observer  ? 

QUERY  2.  How  would  a  horizontal  plane  look  if  the  eye  were  just 
level  with  it? 

QUERY  3.  How  does  the  shape  of  the  top  of  a  level  table  seem  to 
change  if  you  stand  directly  in  front  of  it,  a  few  feet  away,  first  with 
the  eye  level  with  the  top,  and  then  rise  to  your  full  height?  What 
difference  do  you  observe  if  you  repeat  the  process  but  do  not  stand 
directly  in  front  of  the  table  ?  Draw  figures  showing  the  different  forms 
that  the  table  top  presents  from  your  various  positions. 

388.  Coplanar.  Lines  or  points  which  lie  in  the  same  plane 
are  said  to  be  coplanar. 

QUERY  4.    Are  two  parallel  lines  necessarily  coplanar? 

QUERY  5.  If  three  points  are  collinear,  can  more  than  one  plane  be 
found  which  will  contain  all  of  them?  Illustrate.  Do  three  collinear 
points  determine  a  plane? 

QUERY  6.  Do  any  four  points  taken  at  random  in  space  determine  a 
plane  ?  Illustrate. 

QUERY  7.  Why  does  a  three-legged  stool  stand  firmly  on  a  level  floor, 
while  a  four-legged  one  is  likely  to  be  unsteady? 

QUERY  8.  What  does  a  moving  line  generate  if  it  always  passes 
through  a  given  point  and  always  intersects  a  given  line  ? 

QUERY  9.  What  does  a  moving  line  generate  if  it  always  intersects 
two  intersecting  lines? 


BOOK  VI  309 

EXERCISES 

1.  Any  transversal  of  two  parallel  lines  lies  in  the  plane  of 
those  lines. 

2.  If  a  line  cuts  three  concurrent  lines  at  points  other  than 
their  intersection,  the  four  lines  are  coplanar. 

3.  If  a  plane  contains  one  of  two  parallel  lines  and  one  point 
of  the  other,  it  must  contain  both  of  the  parallels. 

HINTS.    Use  §§  385,  384. 

389.  Postulate  VIII.    If  two  planes  have  one  point  in  common, 
they  must  have  at  least  two  points  in  common. 

390.  Intersection.    The  intersection  of  two  lines,  curves,  or 
surfaces  comprises  those  points,  and  only  those,  which  they 
have  in  common. 

Intersections  of  geometric  figures  fall  into 
several  classes,  which  are  defined  as  follows : 

If  two  intersecting  lines,  curves,  or  sur- 
faces pass  through  each  other,  they  are  said 
to  cut  each  other. 

If  one   of   two  intersecting   lines,   curves,   or  surfaces  is 
terminated  at  their  intersection,  it  is  said  to  meet  the  other. 

If  both  of  two  figures  terminate  at  their  in- 
tersection, they  are  said  to  meet  each  other. 

If  a  line  either  cuts  or  meets  a  plane,  the 
intersection  consists  of  only  one  point.  If  a 
line  lies  entirely  in  a  plane,  one  frequently 
says  that  the  plane  passes  through  or  contains  the  line. 

QUERY  1.    Do  the  two  sides  of  an  angle  cut  each  other? 
QUERY  2.   In  the  figure  above  is  it  correct  to  say  that  the  line  meets 
the  plane  or  that  the  plane  meets  the  line  ? 

QUERY  3.    Do  the  two  planes  above  meet  or  cut  each  other? 
QUERY  4.    Hold  two  sheets  of  paper  so  that  they  meet  each  other. 


310 


SOLID  GEOMETRY 
Theorem  2 


391.  If  two  planes   intersect,    their   intersection   is   a 
straight  line. 


Given  two  planes  MN  and  PQ  which  intersect. 
To  prove  that  their  intersection  is  a  straight  line. 

Proof.  MN  and  PQ  must  have  at  least  two  points,  as  A  and  B, 
in  common.  §  389 

Hence  both  MN  and  PQ  contain  the  line  AB  determined  by 
these  points.  §  381 

But  Jl/JVand  PQ  cannot  have  any  point  outside  AB  in  common, 
else  the  planes  would  coincide.  §  384 

Therefore  the  intersection  of  MN  and  PQ  is  a  straight  line. 

QUERY  1.  Can  you  imagine  two  planes  which  do  not  have  ap- 
points in  common  ?  Illustrate. 

QUERY  2.  How  many  planes  can  be  passed,  meeting  a  given  plane 
in  a  given  line  ?  Illustrate. 

QUERY  3.  If  three  planes  in  space  are  taken  at  random,  what  is 
their  intersection  ?  Give  an  example. 

QUERY  4.  What  is  the  least  number  of  planes  that  can  inclose  a 
space  ?  Give  an  example. 

QUERY  5.  What  relations  other  than  the  one  given  in  the  answer  to 
Query  3  may  three  planes  bear  to  each  other  ?  Give  examples. 

QUERY  6.    If  two  surfaces  intersect  in  a  straight  line,  is  it  necessary 
that  both  of  them  be  planes  ?   Give  examples. 
QUERY  7.    Is  the  converse  of  Theorem  2  true  ? 


BOOK  VI  311 

PAEALLEL  LINES  AND  PLANES 

392.  Parallel  planes.    Two  planes  which  do  not  meet  how- 
ever far  they  are  produced  are  said  to  be  parallel.* 

Theorem  3 

393.  If  a  p lane  intersects  two  parallel  planes,  the  inter- 
sections are  parallel  lines. 


Given  the  plane  M  parallel  to  the  plane  N,  and  both  cut  by 
the  plane  Qy  in  lines  a  and  b  respectively. 

To  prove  that  a  is  II  to  b. 

Proof.  a  and  b  are  straight  lines.  §  391 

In  order  to  prove  a  II  to  b,  we  must  show,  first,  that  they  lie  in 
the  same  plane ;  second,  that  they  cannot  meet. 

Now  a  and  b  lie  in  Q.  Given 

Also,  a  and  b  cannot  meet,  since,  if  they  did  meet,  the  planes  M 

and  N  would  have  a  point  in  common,  which  is  impossible.    §  392 

Therefore  a  is  II  to  b.  §  43 

QUERY  1.  A  point,  a  line,  and  a  plane  are  given.  What  is  the  inter- 
section of  the  plane  with  a  moving  line  which  contains  the  given  point 
and  cuts  the  given  line  ?  Mention  any  special  cases  that  may  occur. 

*  In  this  definition  it  is  implied  that  if  planes  meet  they  are  not  parallel. 
A  similar  remark  applies  to  most  of  the  definitions  in  this  and  other  texts. 


312  SOLID  GEOMETRY 

QUERY  2.  If  two  parallel  planes  are  given,  is  it  certain  that  there  is  a 
plane  which  intersects  both  of  them  ?  How  many  such  planes  are  there  ? 
Are  these  planes  necessarily  parallel? 

QUEKY  3.  Where  is  the  figure  of  Theorem  3  situated  with  respect 
to  the  eye  ? 

QUERY  4.  Draw  a  figure  for  Theorem  3  which  appears  to  be  below 
and  to  the  right  of  the  eye. 

394.  Corollary.  Parallel  line-segments  intercepted  between 
parallel  planes  are  equal. 


Given  a  and  &,  parallel  line-segments  intercepted  between  the 
planes  M  and  N. 

To  prove  that  a  is  equal  to  b. 

HINTS.  Pass  the  plane  determined  by  a  and  b  intersecting  M  and  N 
in  x  and  y  respectively. 

Prove  that  a  parallelogram  is  formed. 

QUERY  5.    Is  a  parallelogram  necessarily  a  plane  figure? 

QUERY  6,  Is  every  closed  four-sided  figure  necessarily  a  plane  figure  ? 
Illustrate. 

QUERY  7.  If  two  parallel  planes  cut  off  equal  lengths  on  two  lines, 
are  the  lines  necessarily  parallel  ?  Illustrate. 

QUERY  8.  If  two  parallel  planes  cut  off  equal  lengths  on  two  lines, 
do  the  lines  necessarily  intersect  if  sufficiently  produced  ?  Illustrate. 

EXERCISE  4.  Given  two  parallel  planes  which  cut  off  equal 
segments  on  two  intersecting  lines.  Pass  the  plane  of  the  inter- 
secting lines  and  show  that  two  isosceles  triangles  are  formed. 


BOOK  VI.  313 

395.  Parallel  lines  and  planes.    A  line  and  a  plane  that  do 
not  meet  however  far  they  are  produced  are  said  to  be  parallel. 

QUERY  1.  If  a  line  is  parallel  to  two  planes,  are  the  planes  necessarily 
parallel  to  each  other  ?  Illustrate. 

QUERY  2.  If  two  planes  are  parallel,  is  a  given  line  in  one  plane 
parallel  to  every  line  in  the  other  ?  Is  it  parallel  to  some  line  in  the 
other? 

QUERY  3.  How  many  lines  are  there  through  a  given  point  parallel 
to  a  given  plane  ?  Illustrate. 

QUERY  4.  In  what  kind  of  surface  do  you  think  all  the  lines  of 
Query  3  would  lie? 

QUERY  5.  Hold  a  pointer  so  that  it  is  parallel  to  a  side  and  an  end 
wall  of  the  room.  How  many  straight  lines  are  there  through  a  given 
point  parallel  to  each  of  two  intersecting  planes  ? 

Theorem  4 

396.  If  a  plane  contains  only  one  of  two  parallel  lines, 
it  is  parallel  to  the  other  line. 


Given  the  line  a  parallel  to  the  line  &,  and  the  plane  M  con- 
taining b  hut  not  containing  a. 

To  prove  that  M  is  II  to  a. 

Proof,    a  meets  M,  if  at  all,  in  some  point  X  which  is   not 

on  b.  Why  ? 

Through  X  draw  c  in  M  II  to  b.  §  43 

Then  we  have  through  X  two  lines,  a  and  c,  both  II  to  b, 

which  is  impossible.  §  45 

Hence  a  cannot  meet  M  and  is  II  to  it.  §  395 


314 


SOLID  GEOMETRY 


397.  Constructions.    For  the  present  a  construction  in  solid 
geometry  means  the  building  of  a  figure  by  application  of 
the  following  actual  or  imagined  operations: 

1.  The  passing  of  planes  (§§  382,  383,  384,  385). 

2.  The    determination    of    lines    by    the    intersection    of 
planes  (§  391). 

3.  The  use  of  ruler  and  compasses  in  planes. 

The  third  operation  refers  to  the  constructions  of  plane  geome- 
try performed  in  the  planes  afforded  by  the  first  process.  Instead 
of  having  only  one  method  of  determining  a  line,  as  was  the  case 
in  plane  geometry,  we  now  have  two :  a  pair  of  points  and  a  pair 
of  nonparallel  planes. 

Construction  1 

398.  Through  a  point  outside  a  plane  construct  a  line 

to  the  plane. 


Given  the  point  P  outside  the  plane  M. 
Required  to  construct  a  line  through  P  II  to  M. 

Construction.    Through  the  point  P  pass  a  plane  N  intersecting 
M  in  some  line,  as  a. 

In  the  plane  N  draw  a  line  b  through  P  II  to  a.  §  45 
Then                                   b  is  II  to  M. 

Proof.                                    b  is  II  to  a.  Const. 

Therefore                           I  is  II  to  M.  §  396 


BOOK  VI  315 

Theorem  5 

399.  If  a  line  is  parallel  to  a  plane,  the  intersection  of 
the  j)lanv  with  a  plane  passed  through  the  line  is  parallel 

to  the  line. 

A B 


Given  the  line  AB  parallel  to  the  plane  M\  and  the  plane 
AL  containing  AB  and  intersecting  M  in  KL. 

To  prove  that  AB  is  II  to  KL. 

Proof  is  left  to  the  student. 

QUERY  1.    To  how  many  lines  in  a  given  plane  may  a  line  be  parallel  ? 

QUERY  2.  Under  what  conditions  is  a  straight  stick  parallel  to  its 
own  shadow  on  the  ground? 

QUERY  3.  Under  what  conditions  may  a  line  be  parallel  to  each  of 
three  planes?  Illustrate. 

QUERY  4.  How  many  planes  are  there  through  a  given  point  parallel 
to  a  given  line  ?  Illustrate.  .  • 

EXERCISES 

5.  If  a  line  and  a  plane  are  parallel,  a  line  containing  a  point 
of  the  plane  and  parallel  to  the  given  line  lies  wholly  in  the  plane. 

HINTS.  Let  AB  be  parallel  to  M  and  let 
PR  meet  M  in  P  and  be  parallel  to  AB. 
Pass  a  plane  determined  by  AB  and  PR, 
meeting  plane  M  in  PQ.  Show  that  PQ  and 
PR  are  both  II  to  AB. 


6.  Through  a  given  point  construct 
a  line  parallel  to  a  given  plane  and 
meeting  a  given  line.  Is  this  construction  always  possible  ? 


316 


SOLID  GEOMETRY 


7.  Two  intersecting  planes  are  each  parallel  to  a  given  line. 
What  is  the  relation  between  the  intersection  of  these  planes  and 
the  line  ?  Prove  your  statement. 

HINT.  Pass  a  plane  determined  by  the  given  line  and  any  point 
of  the  intersection  of  the  planes. 


Theorem  6 


400.  If  two  intersecting  lines  are  parallel  to  a  plane> 
their  plane  is  parallel  to  the  plane. 


Given  the  lines  a  and  &,  both  parallel  to  the  plane  M\  and  Q, 
their  plane. 

To  prove  that  Q  is  II  to  M. 

Proof.  If  Q  should  intersect  M  in  a  line  x,  then  a  and  b  would 
each  be  II  to  x.  §  399 

We  should  then  have  through  the  point  P  two  lines,  each  II  to 
the  same  line,  which  is  impossible.  §  45 

Therefore  Q  does  not  meet  M,  and  is  II  to  it.  Why? 

401.  Corollary.  If  two  intersecting 
lines  are  respectively  parallel  to,  but  not 
coplanar  with,  two  other  intersecting  lines, 
the  plane  of  the  first  pair  is  parallel  to 
the  plane  of  the  second  pair. 

HINTS.    Let  N  be  determined  by  c  and  d. 
Then  a  and  b  are  each  II  to  ^V  by  §  396. 


BOOK  VI 


317 


Theorem  7 

402.  If  a  plane  intersects  one  of  two  parallel  lines,  it 
intersects  the  other  also. 


10  R 


Given  the  line  a  parallel  to  the  line  &,  and  the  plane  M  in- 
tersecting b  Sit  the  point  0. 

To  prove  that  M  also  intersects  a. 

Proof.    Pass  the  plane  N  determined  by  a  and  I,  intersecting 
M  in  RP.  §  385 

Then  RP  must  intersect  a.  §  46 

Hence    M,  the  plane  in  which  RP  lies,  must  intersect  a. 

NOTE.  It  is  always  desirable  to. observe  whether  the  reason  for  a 
given  step  is  taken  from  plane  geometry, 
and,  if  so,  to  note  the  plane  containing  the 
figure  to  which  the  reference  is  made.  In 
the  foregoing  proof  the  figure  to  which  §  46 
applies  lies  in  the  plane  of  a  and  b. 

4C3.  Corollary  1.  If  a  line  intersects 
one  of  two  parallel  planes,  it  intersects 
the  other  also. 

HINTS.  Pass  a  plane  determined  by  a  and  any 
point  P  of  M.  Apply  §  46. 

404.  Corollary  2.  If  a  plane  intersects  one 
of  two  parallel  planes,  it  intersects  the  other 
also. 

HINT.  .  In  the  cutting  plane  draw  a  line  cutting 
the  line  of  intersection  of  the  two  planes. 


318 


SOLID  GEOMETRY 


405.  Corollary  III.    If  tiuo  planes  are  par- 
allel  to   the   same  plane   they  are  parallel   to 

each  other.  /K / 

HINT.    Let  M  be  II  to  N  and  to  P.    If  N  should      yjT~  ~~7 

intersect  P,  show  that  it  would  also  intersect  M. 

Theorem  8 

406.  If  two  lines  are  parallel  to  the  same  line,  they  are 
parallel  to  each  other. 


Given  the  lines  b  and  c  each  parallel  to  a. 

To  prove  b  and  c  parallel  to  each  other. 

Proof.    Pass  the  plane  M  containing  I  and  one  point  of  c.  §  384 

Now  M  either  cuts  c  or  contains  it. 

If  M  cuts  c,  it  will  cut  a  and  b.  §  402 

But  M  cannot  cut  b  for  it  contains  b.  Const. 

Hence  M  cannot  cut  c.    Therefore  M  contains  c,  since  the  only 
other  possibility  leads  to  a  contradiction. 

Therefore  b  and  c  lie  in  the  same  plane. 

Since  they  are  each  II  to  a,  Given 

b  cannot  meet  c.  §  45 

Hence                                   b  is  II  to  c. .  §  43 

NOTE.    It  should  be  observed  that  §  45  is  assumed  to  be  true  in 
space  as  well  as  in  a  plane. 


BOOK  VI       .  319 

Theorem  9 

407.  If  two  angles  not  in  the  same  plane  have  their  sides 
parallel  and  extending  in  the  same  direction  from  their 
vertices,  they  are  equal. 

-B 


Given  the  angles  LAK  and  RGS  in  the  planes  M  and  N 
respectively,  with  sides  AL  and  AK  parallel  respectively  to 
sides  GR  and  GS. 


To  prove  that 

Proof.  Construct  A  B  =  GF,  and  A  C  =  Gil,  and  draw  A  G,  CH, 
and  1>F. 

Now  EG  and  CG  are  parallelograms.  §  88 

Hence  BF  and  CH  are  each  II  to  A  G.  Why  ? 

Consequently                  BF  is  II  to  CH.  §  406 

Also                                      BF  =  CIL  Why  ? 

Hence  the  figure  BCIIF  is  a  parallelogram,  §  88 

and  CB  =  HF.  Why? 

In  ABA  C  and  FGH,  BA  =  FG  and  4  C  =  GH,  Const. 

and  we  have  proved  CB  =  HF. 

Therefore  the  A  are  congruent,  Why? 

and  Z.BAC  =  Z.FGH.  Why? 


QUERY  1.  If  the  angles  of  Theorem  9  have  their  sides  parallel  each 
to  each,  but  extend  in  opposite  directions  from  their  vertices,  what  is 
the  relation  between  the  angles  ? 


320  SOLID  GEOMETRY 

Construction  2 

408.   Through  a  point  not  in  a  plane,  construct  a  plane 
parallel  to  that  plane. 


57 


7 


Given  the  point  P  and  the  plane  M. 

Required  to  construct  a  plane  containing  P  and  parallel  to  M. 

Construction.    Construct  PR  and  PK  II  to  M.  §  398 

Pass  plane  N  determined  by  PR  and  PK. 

Hence  N  is  II  to  M. 

Proof.    Since  PR  and  PK  are  each  li  to  M,  N  is  II  to  M.      §  400 

Theorem  10 

409.   Through  a  point  not  lying  in  a  plane,  one  and 
only  one  plane  can  be  passed  parallel  to  that  plane. 


7 


Given  the  point  P  and  the  plane  M. 

To  prove  that  one  and  only  one  plane  through  P  is  II  to  M. 


BOOK  VI  321 

Proof.    Denote  by  N  the  plane  constructed  II  to  M  and  con- 
taining P.  §  408 
Any  plane  through  P  other  than  N,  such  as  R,  would  cut  N.   §  391 

Therefore  R  would  cut  M,  §  404 

which  is  contrary  to  the  hypothesis. 
Hence         N  is  the  only  plane  through  P  II  to  M. 

QUERY  1.  How  many  planes  are  there  parallel  to  two  given  parallel 
planes  ?  Illustrate. 

QUERY  2.  How  many  planes  are  there  through  a  given  point  parallel 
to  two  given  parallel  planes  ?  Illustrate. 

QUERY  3.  Two  lines  are  given.  If  a  moving  line  is  always  parallel  to 
one  of  the  given  lines  and  always  intersects  the  other,  what  surface  is 
generated  and  what  "is  its  position? 

QUERY  4.  In  the  statement  "Through  a  given  point  outside  a 

one parallel  to  it  can  be  drawn,  and  only  one,"  fill  the  blank  spaces 

with  the  words  line  and  plane  in  each  of  the  four  possible  ways.  Which 
of  the  resulting  statements  are  true  ?  In  what  section  is  each  proved 
or  assumed  ? 

QUERY  5.  If  the  angles  of  Theorem  9  have  their  sides  parallel  each 
to  each,  and  if  one  pair  extend  in  the  same  direction  from  the  vertices 
while  the  other  pair  extend  in  opposite  directions,  what  is  the  relation 
between  the  angles? 

EXERCISES 

8.  If  two  lines  are  not  in  the  same  plane,  one  plane  and  only 
one  can  be  passed  containing  one  of  these  lines  and  parallel  to 
the  other. 

9.  Construct  a  line  through  a  given  point  parallel  to  two  given 
intersecting  planes. 

HINT.  Pass  the  plane  determined  by  the  point  and  the  intersection 
of  the  planes. 

10.  If  one  of  two  parallel  lines  is  parallel  to  a  plane,  the  other 
is  also. 

11.  Through  a  given  point  one  and  only  one  plane  can  be 
passed  parallel  to  two  given  nonparallel  lines  in  space. 


322 


SOLID  GEOMETRY 


Theorem  11 

410.  If  two  straight  lines  are  cut  by  three  parallel  planes, 
the  corresponding  segments  are  proportional. 


Given  three  parallel  planes  M,  N,  R,  cutting  two  lines  AD 
and  CD  in  the  points  -A,  F,  B  and  C,  H,  D  respectively. 

To  prove  that  |f=||- 

Proof.    Draw  AD;  pass  the  planes  of  DC  and  DA,  and  of  AB 
and  AD.    Denote  the  intersections  with  the  given  planes  by  AC, 

GH,  FG,  BD. 

§393 


In  AADC. 
In  AABD, 
Therefore 


AC  is  II  to  Gil,  and  FG  is  II  to  BD. 

CH  _AG 
II D  ~  GD ' 

AF  _AG 
FB  ~  GD' 

AF  _  CH 
~FB~  HD' 


Why? 
Why  ? 
Why? 


411.   Corollary.    If  two  parallel  planes  cut  a  series  of  con- 
current lines,  the  corresponding  segments  are  proportional. 

HINT.    Pass  the  plane  through  the  point  common  to  the  lines  and 
parallel  to  one  of  the  given  planes  (§  409). 


BOOK  VI  323 

QUERY  1.  In  what  case  are  F,  G,  and  //  of  Theorem  11  in  a  straight 
line  ?  Under  what  conditions  are  A  C  and  BD  parallel  ? 

QUERY  2.  Describe  the  positions  of  the  planes  that  are  drawn 
through  a  fixed  point  so  as  to  contain  a  set  of  parallel  coplanar  lines ; 
of  parallel  lines,  not  all  coplanar;  of  concurrent  coplanar  lines;  of 
concurrent  lines,  not  all  coplanar. 

REVIEW  EXERCISES 

12.  A  quadrilateral  is  a  plane  figure  if  two  of  its  sides  are 
parallel. 

13.  If  any  number  of  parallel  lines  meet  a  given  line,  they  are 
all  coplanar. 

14.  If  each  of  three  lines  meets  the  other  two,  the  three  are 
either  coplanar  or  concurrent. 

15.  If  two  lines  are  not  coplanar,  show  that  it  is  impossible  to 
draw  two  parallel  lines  each  cutting  both  the  given  lines. 

16.  Given  two  lines  which  do  not  meet  and  are  not  parallel. 
Through  a  given  point  construct  a  third  line  meeting  both  the 
given  lines.    Discuss  the  special  cases. 

17.  If  the  foot  of  a  ten-foot  pole  is  placed  on  the  bottom  of  a 
body  of  water  8  feet  deep,  and  the  top  of  the  pole  is  at  the  sur- 
face, will  the  middle  of  the  pole  always  lie  in  the  same  plane  ? 
Prove  your  statement. 

18.  If  a  line  in  one  of  two  intersecting  planes  is  parallel  to  a 
line  in  the  other,  both  lines  are  parallel  to  the  intersection  of  the 
planes. 

19.  Given  four  lines  in  space,  only  two  of  which  are  parallel. 
Construct  a  line  cutting  all  four  lines.    Discuss  the  special  cases. 

20.  The  top  of  a  ten-foot  pole  is  placed  in  the  corner  of  a  room 
at  the  ceiling.    The  foot  of  the  pole  is  found  to  be  on  the  floor 
6  feet  from  the  corner.    How  high  is  the  room  ? 

21.  If  three  parallel   planes  cut  off  equal  segments  on  one 
transversal,  they  cut  off  equal  segments  on  every  transversal. 


324  SOLID  GEOMETRY 

22.  In  one  of  two  parallel  planes  three  lines  are  drawn  which 
are  parallel,  each  to  each,  to  lines  in  the  other  plane.    Are  the 
triangles  formed  in  the  two  planes  necessarily  similar?    Prove 
your  statement.    Discuss  special  cases. 

23.  To  which  of  the  preceding  theorems  is  the  following  state- 
ment equivalent  ?    "  If  a  line  is  parallel  to  a  line  in  a  plane,  it  is 
either  parallel  to  or  contained  by  the  plane." 

24.  Construct  a  line  parallel  to  a  given  plane  and  meeting  each 
of  two  given  lines.    Discuss  special  cases. 

25.  Construct  a  line  which  cuts  three  given  lines.    Is  more 
than  one  such  line  possible  ?   Discuss  special  cases. 

26.  Through  a  given  point  pass  two  planes,  one  parallel  to  each 
of  two  given  intersecting  planes.    What  can  you  say  of  the  inter- 
section of  the  two  planes  so  drawn  ?   Prove  your  statement. 

27.  Construct  a  plane  which  shall  pass  through  a  given  line 
and  cut  two  given  planes  in  parallel  lines. 

PERPENDICULAR  LINES  AND  PLANES 

412.  Foot  of  a  line.  The  point  where  a  line  intersects  a 
plane  is  called  the  foot  of  the  line. 

QUERY  1.  Can  a  line  have  two  feet  in  a  given  plane  and  at  the  same 
time  cut  the  plane  ? 

QUERY  2.  Are  there  any  lines  which  have  no  foot  in  a  given  plane  ? 
Illustrate. 

QUERY  3.  At  a  given  point  in  a  line,  how  many  perpendiculars  to 
the  line  are  there? 

QUERY  4.  How  are  the  planes  arranged  which  are  determined  by  a 
given  line  and  the  various  perpendiculars  at  a  point  on  the  line? 

QUERY  5.  If  two  lines  are  perpendicular  to  the  same  line,  are  they 
necessarily  parallel?  Illustrate. 

QUERY  6.  Keeping  in  mind  that  one  must  always  provide  a  plane 
in  space  in  which  to  perform  a  construction  of  plane  geometry,  how 
would  you  construct  two  lines  perpendicular  to  a  given  line  at  the 
same  point? 


BOOK  VI 


325 


Theorem  12 

413.  If  a  line  is  perpendicular  to  two  lines  at  their 
point  of  intersection,  it  is  perpendicular  to  any  line  in 
their  plane  through  that  point. 


Given  OX  and  OF,  two  lines  in  the  plane  Af,  each  perpen- 
dicular to  OP  at  0,  and  let  OZ  be  any  other  line  through  0  in  M. 

To  prove  that  OP  is  _L  to  OZ. 

Proof.  Draw  any  line  in  M  not  through  0,  cutting  OX,  0  Y,  and  OZ 
at  A,  B,  and  C  respectively.  Produce  PO  to  A",  making  OK  =  OP. 
Draw  lines  PA,  PB,  PC,  KA,  KB,  and  KG. 

In  the  plane  determined  by  PK  and  OA,  PA  =  KA. 

In  the  plane  determined  by  PK  and  OB,  PB  =  KB. 

Therefore          &PAB  and  KAB  are  congruent. 

Hence  Z  PB  C  =  Z  KB  C. 

Also  CB  =  CB. 

Therefore  APBC  =  AKBC. 

Consequently,  KC  =PC. 


§116 
Why? 

Why? 
Why? 

Why? 

§27 


Hence  OZ  contains  two  points,  0  and  C,  equidistant  from  P 
and  K,  and  is  therefore  _L  to  PK  at  0.  §  118 

Therefore  OP  is  _L  to  OZ. 

414.  Perpendicular  to  a  plane.  A  line  is  perpendicular  to 
a  plane  if  it  is  perpendicular  to  every  line  in  the  plane 
drawn  through  its  foot. 


326  SOLID  GEOMETRY 

If  a  line  intersects  a  plane  and  is  not  perpendicular  to  it,  it  is 
said  to  be  oblique  to  the  plane. 

From  this  definition  it  appears  that  if  a  line  makes  an  oblique 
angle  with  any  line  of  a  plane,  it  cannot  be  perpendicular  to  the 
plane.  But  one  cannot  show  directly  from  the  definition  that  a  line 
is  perpendicular  to  a  plane  without  testing  the  angle  which  is  found 
with  every  line  through  its  foot,  —  a  process  which  could  never  be 
completed,  since  it  would  require  an  infinite  number  of  operations. 

The  great  importance  and  power  of  Theorem  12  consists  in 
two  facts :  first,  it  shows  that  a  line  can  be  perpendicular  to  all 
of  the  lines  in  a  plane  through  its  foot ;  second,  it  replaces  the 
infinite  number  of  operations  mentioned  above  by  only  two,  mak- 
ing it  possible  to  show  that  a  line  is  perpendicular  to  all  lines  in 
the  plane  drawn  through  its  foot  if  it  is  found  to  be  perpendicular 
to  just  two  of  them. 

415.   Corollary.     If  a  line   is  perpendicular  to  two   lines  at 
their  point  of  intersection,  it  is  perpendicular  to  their  plane. 
This  follows  immediately  from  §§  413,  414. 

QUERY  1.  If  one  line  is  perpendicular  to  another,  is  a  plane  con- 
taining the  first  line  sure  to  be  perpendicular  to  the  second  ? 

QUERY  2.  How  could  you  determine  by  use  of  a  carpenter's  square 
whether  a  square  post  is  perpendicular  to  a  level  floor?  How  many 
operations  are  necessary? 

QUERY  3.  In  the  figure  for  Theorem  12,  which  point  is  nearer  the 
eye,  A  or  0?  B  or  (9?  C  or  0? 

QUERY  4.    Where  is  the  triangle  OAB  with  respect  to  the  eye? 

QUERY  5.  Stand  a  pencil  on  end  on  a  sheet  of  paper  which  lies  on 
a  level  table.  Draw  several  lines  on  the  paper  through  the  end  of  the 
pencil.  Hold  up  a  book  so  that  one  corner  is  between  the  eye  and  the 
point  where  the  pencil  meets  the  paper,  and  one  edge  of  the  book  is  in 
line  with  the  pencil.  In  this  way  test  which  of  the  right  angles  formed 
should  appear  as  right  angles  in  a  drawing. 

QUERY  6.  May  a  right  angle  ever  be  correctly  represented  by  an 
obtuse  angle?  an  acute  angle?  a  straight  angle?  an  angle  of  zero 
degrees  ?  Explain. 

QUERY  7.  Can  two  pointers  at  right  angles  be  held  in  such  a  position 
that  the  angle  which  they  form  appears  to  the  class  to  be  obtuse  ? 


BOOK  VI 


327 


Construction  3 

416.   Construct  a  plane  containing  a  given  point  and 
perpendicular  to  a  given  line. 


Given  the  point  P  and  the  line  a. 

Required  to  construct  a  plane  containing  P  and  _L  to  a. 

Case  I.    When  P  is  on  the  line  a. 

Construction.    At  P   construct   two   lines,    PR   and   PS,   each 

§234 
§382 


_L  to  a. 

Pass  the  plane  M  determined  by  PR  and  PS. 

M  is  _L  to  a  at  P. 

Proof.  a  is  _L  to  PR  and  to  PS. 

Therefore  a  is  J_  to  M. 

Case  II.    When  P  is  not  on  the  line  a. 


Const. 
§415 


Construction.    From  P  drop  a  _L  to  a  and  denote  the  intersection 
by  S.  §  235 

At  S  draw  ST,  another  _L  to  a.  §  234 

Pass  the  plane  M  determined  by 
SP  and  STY  §382 


M  is  _L  to  a. 
Proof  is  left  to  the  student. 


328 


SOLID  GEOMETRY 
Theorem  13 


417.   One  and  only  one  plane  can  be  passed  containing 
a  given  point  and  perpendicular  to  a  given  line. 


K 


B 


Given  the  point  P  and  the  line  AB. 

To  prove  that  one   and  only  one  plane  can  be  passed  con- 
taining P  and  _1_  to  AB. 

Case  I.    When  P  is  on  AB. 

Proof.    Denote  by  Afa  plane  _L  to  AB  at  P.  §416 

Suppose  there  were  another  plane,  as  KL,  also  _L  to  AB  at  P. 
Let  PT  be  any  line  in  M  through  P. 

Pass  the  plane  A  T  determined  by  P77and  AB,  cutting  KL  in 
the  line  PS. 

Then  PT  and  PS  are  both  _L  to  AB  at  P.  §  414 

But  PT  and  PS  are  both  in  the  plane  A  T. 

Therefore  PT  and  PS  must  coincide. 


But  P  T  is  drawn  as  any  line  in  M  through  P. 
Hence  M  coincides  with  KL. 

Therefore  only  one  plane  can  be  _L  to  AB  at  P. 


§41 


382 


QUERY.  If  a  line  a  meets  a  plane  M,to  which  it  is  not  perpendicular, 
at  a  point  /*,  does  a  line  exist  in  M  through  P  to  which  a  is  perpen- 
dicular? Does  more  than  one  such  line  exist? 


BOOK  VI 


829 


Case  II.    When  P  is  not  on  AB. 

Proof.    Denote  by  M  a  plane  containing  P  and  _L  to  AB.    Sup- 
pose there  were  another  plane,  N,  containing  P  and  also  _L  to  AB. 

Now  M  and  N  could  not 
both  intersect  AB  at  the  same 
point.  Case  I 

Let  T  and  S  be  the  intersec- 
tions of  M  and  N  respectively 
with  AB. 

Then  PT  and  PS  are  both  in 
the  plane  determined  by  A  B  and 
P,  and  are  both  _L  to  AB.  §414 

Therefore  PT  and  PS  must  coincide, 

and  M  coincides  with  N. 


§42 
Case  I 


Hence  only  one  plane  can  be  _L  to  AB  through  P. 


418.  Corollary.    All  of  the  lines  perpendicular  to  a  line  at 
a  point  lie  in  a  plane  perpendicular  to  that  line  at  that  point. 

HINTS.  "Let  BK  and  BR  be  any  two  Js  to  A  B.  Pass  plane  MN  through 
BK  and  BR.  Let  EL  be  any  other  JL  to 
AB  at  B.  Pass  plane  through  BL  and  BK 
and  show  that  the  two  planes  coincide. 

QUERY  1.   In  order  to  prove  that  the 
plane  perpendicular  to  a  line  at  a  given 
point  is  the  locus  of  lines  perpendicular 
to  the  given  line  at  that  point,  what  two  facts  must  be  established? 
Are  sections  414  and  418  sufficient  for  this  purpose  ? 

QUERY  2.    If  a  right  angle  be  rotated  about  one  of  its  sides,  what 
does  the  other  side  generate  ? 

QUERY  3.    What  kind  of  surface  does  a  spoke  of  a  wheel,  which  is 
not  dished,  generate  in  its  rotation  ? 

QUERY  4.    Can  a  plane  always  be  passed  parallel  to  one  of  two  lines 
in  space  and  perpendicular  to  another  ? 

EXERCISE  28.    If  two  planes  are  perpendicular  to  the  same  line, 
the  planes  are  parallel. 


830 


SOLID  GEOMETRY 
Theorem  14 


419.  If  a  line  is  perpendicular  to  one  of  two  parallel 
planes,  it  is  perpendicular  to  the  other  also. 


Given  the  parallel  planes  M  and  N  and  the  line  AB  perpen- 
dicular to  the  plane  M  at  the  point  0. 

To  prove  that  AB  is  A.  to  N. 

Proof.  A  B  intersects  N  in  some  point  S.  §  403 

Through  the  point  S  draw  two  lines  in  the  plane  JV,  as  SD 
and  SF. 

Pass  the  planes  SH  and  SG,  determined  by  SD  and  AB,  and  by 
SFand  AB  respectively. 

Let  the  intersections  of  these  planes  with  M  be  called  OH  and 
OG  respectively. 

OG  is  II  to  SF,  and  OH  is  II  to  SD.  §  393 

But                        AB  is  _L  to  OH  and  to  OG.  Why? 

Hence                    AB  is  _L  to  SD  and  to  SF.  §  47 

Therefore                          AB  is  _L  to  N.  Why? 

QUERY  1.  Will  a  pointer  held  perpendicular  to  the  floor  of  a  room 
also  be  perpendicular  to  the  ceiling? 

QUERY  2.  To  how  many  planes  is  a  given  line  perpendicular? 
Illustrate. 


BOOK  VI 


331 


Construction  4 

420.   Construct  a  line  perpendicular  to  a  given  plane 
and  containing  a  given  point. 


Given  the  point  P  and  the  plane  M. 

Required  to  construct  a  line  containing  P  and  _L  to  M. 

Case  I.    When  P  lies  in  M. 

Construction.    Draw  any  line  PA  in  M  through  P,  and  pass  the 


plane  KL  J_  to  PA  at  P. 

In  KL  draw  PC  _L  to  the  intersection  PL  at  P 

PC  is  _L  to  M. 
Proof.  PC  is  J_  to  PL. 

PC  is  _L  to  A  P. 
Therefore  PC  is  _L  to  M. 


417 


Const. 
§414 
§415 


Case  II.    When  P  does  not  lie  in  M. 

Construction.    Construct  the  plane  N 
containing  P  and  II  to  M. 


At  P  construct  PR  _L  to  N. 
Then          PR  is  _L  to  M. 
Proof.         PR  is  _L  to  N. 
Therefore  PR  is  _L  to  M. 


§408 
Case  I 

Const. 
§419 


QUERY.    Why  could  not  Case  II  be  proved  as  follows:  "Draw  any 
line  in  M,  and  drop  a  perpendicular  from  P  to  this  line." 


332  SOLID  GEOMETRY 

Theorem  15 

421.  One  and  only  one  line  can  be  drawn  perpendicular 
to  a  given  plane  and  containing  a  given  point. 

C  X 


Given  the  plane  M  and  the  point  P. 

To  prove  that  one  and  only  one  line  can  be  drawn  JL  to  M 
and  containing  P. 

Case  I.    When  P  lies  in  M. 

HINTS.  Denote  by  CP  a  line  containing  P  ±  to  M  (§  420).  If 
possible,  let  PX  also  be  JL  to  M  at  P.  Pass  the  plane  of  CP  and  XP, 
and  show  that  CP  and  XP  coincide.  p 

Case  II.    When  P  does  not  lie  in  M. 

The  proof,  which  is  left  to  the  student, 
may  follow  the  hints  for  Case  I,  but 
referred  to  the  adjacent  figure. 


/ 


422.   Corollary.  The  perpendicular  is  the  shortest  line  that  can 
be  drawn  from  a  point  to  a  plane. 

Given  (see  figure  for  Case  II  above)  the  plane  My  and  PC 
the  perpendicular  to  M  from  P. 

To  prove  that  PC  is  the  shortest  line  from  P  to  the  plane  M. 
Let  PX  be  any  other  line  from  P  to  M. 

In  A  PCX,  Z  C  is  a  right  angle.  §  414 

Hence  PC  is  shorter  than  P.Y.  §  149 

QUERY.    Is  the  shortest  distance  from  a  point  to  a  given  line  in  a 
plane  necessarily  the  shortest  distance  from  the  point  to  the  plane  ? 


BOOK  VI  333 

Theorem  16 

423.  The  locus  of  points  in  space  which  are  equidistant 
from  two  given  points  is  the  plane  which  bisects  perpen- 
dicularly the  line-segment  joining  the  points. 


Given  two  points,  A  and  5,  and  the  plane  M,  which  is  perpen- 
dicular to  the  line  AB  at  its  middle  point  R. 

To  prove  that  M  is  the  locus  of  points  in  space  equidistant  from 
A  and  B;  or  (1)  that  every  point  in  Mis  equidistant  from  A  and 
B,  and  (2)  that  every  point  in  space  which  is  equidistant  from 
A  and  B  lies  in  M. 

Proof.    (1)  Let  P  be  any  point  in  M.    Draw  PA,  PB,  and  PR. 
A  PAR  is  congruent  to  APBR.  Why  ? 

Therefore  PA  =  PB.  Why  ? 

(2)  Let  K  be  any  point  such  that  KA  =  KB.    Draw  KR. 

AKAR  is  congruent  to  AKBR.  WThy  ? 

Therefore  Z  KRA  =  Z  KRB,  Why  ? 

and  KR  is  _L  to  AB.  §  37 

Hence  KR  lies  in  the  plane  _L  to  AB  at  R,  §  418 

and  consequently  K  lies  in  that  plane. 

Therefore  M  is  the  locus  of  points  in  space  equidistant  from 
A  and  B. 


334  SOLID  GEOMETRY 

QUERY  1.  What  is  the  locus  of  points  which  lie  in  a  given  plane  and 
which  are  equidistant  from  two  points  not  in  that  plane  ? 

QUERY  2.  What  is  the  locus  of  points  equidistant  from  three  given 
points  which  are  not  in  the  same  straight  line  ? 

QUERY  3.  Determine  a  point  in  a  given  plane  which  is  equidistant 
from  three  points  in  space.  Discuss  the  various  special  cases. 

424.  Logical  relation  between  propositions.  If  the  hypothesis 
and  the  conclusion  of  a  proposition  are  interchanged,  the  resulting 
proposition  is  called  the  converse  of  the  original  one.  The  relation 
between  a  proposition  and  its  converse  may  be  expressed  in  terms 
of  symbols  as  follows  : 

(1)  Direct  proposition  :  If  A  is  B,  then  C  is  D. 

(2)  Converse  proposition :  If  C  is  7),  then  A  is  B. 

If  the  negative  of  both  the  hypothesis  and  the  conclusion  is 
taken,  the  resulting  proposition  is  called  the  opposite  of  the 
original.  Using  the  same  notation  as  above, 

(3)  Opposite  proposition  :  If  A  is  not  B,  then  C  is  not  D. 

If  a  direct  proposition  and  its  converse  are  true,  then  the  oppo- 
site proposition  is  true.  Let  us  assume  the  truth  of  (1)  and  of 
(2)  and  prove  that  the  truth  of  (3)  follows.  Now  if  A  is  not  B,  it 
follows  that  C  is  not  D.  For  if  C  were  D,  then  A  would  be  B,  by  (2). 
But  this  contradicts  the  hypothesis  of  (3).  Hence  (3)  is  true. 

In  proving  a  locus  theorem  it  is  sufficient  to  prove  a  propo- 
sition and  its  converse  (cf.  §  423).  In  this  text  locus  theorems 
are  established  by  this  method.  From  the  preceding  discussion 
it  follows  that  another  theorem,  namely,  the  opposite,  follows 
immediately  from  the  proof  of  a  theorem  and  its  converse.  For 
example,  from  Theorem  16  it  follows  that  if  a  point  is  not  in  the 
plane  which  bisects  the  line-segment  joining  A  and  B,  it  is  not 
equidistant  from  A  and  B. 

QUERY  4.  State  the  opposite  of  Theorem  5.    Is  it  true? 

QUERY  5.  State  the  opposite  of'(o)  §413;  (b)  §407;  (c)  §400. 

QUERY  6.  Is  the  opposite  of  a  true  proposition  necessarily  true? 
Illustrate. 

EXERCISE  29.  Prove  that  if  a  proposition  and  its  opposite  are 
true,  then  the  converse  is  true. 


BOOK  VI  335 

Theorem  17 

425.  (1)  If  equal  oblique  line-segments  are  drawn  from 
a  point  to  a  plane,  they  meet  the  plane  at  equal  distances 
from  the  foot  of  the  perpendicular  to  the  plane  from 
that  point.  (2)  If  oblique  line-segments  from  a  point  to  a 
plane  meet  the  plane  at  equal  distances  from  the  foot  of 
the  perpendicular,  the  line-segments  are  equal. 


A 


/AV 

/M     B     / 


(1)  Given  PT  perpendicular  to  the  plane  Af,  and  PA  equal  toPB. 
To  prove  that  TA  =  TB. 

HINT.    Compare  the  &PTA  and  PTB. 

(2)  Given  PT  perpendicular  to  AT,  and  TA  equal  to  TB. 
To  prove  that  PA  =  PB. 

Proof  is  left  to  the  student. 

QUERY  1.  What  is  the  locus  of  points  in  space  equidistant  from  all 
the  points  of  a  given  circle  ? 

QUERY  2.  What  is  the  least  number  of  rigid  iron  braces  that 
will  hold  a  pole  in  vertical  position?  What  is  the  least  number  of 
guy  ropes? 

QUERY  3.  The  ceiling  of  a  room  is  10  feet  high.  How  would  you 
determine  by  means  of  a  12-foot  pole  and  a  pair  of  compasses  a  point 
in  the  floor  directly  under  a  given  point  in  the  ceiling  ? 

QUERY  4.  What  is  the  logical  relation  of  the  proposition  in  §425 
(1)  to  that  in  (2)  ? 

QUERY  5.    State  the  opposite  of  §  425  (1).    Is  it  true?  Why? 


336  SOLID  GEOMETRY 

QUERY  6.    State  the  opposite  of  §425  (2).    Is  it  true?  Why? 
QUERY  7.    What  is  the  locus  of  the  points  in  a  plane  a  fixed  distance 
from  a  given  point  not  in  the  plane  ? 

EXERCISES 

30.  If  two  oblique  lines  drawn  from  a  point  in  a  perpendicular 
to  a  plane  cut  off  unequal  distances  from  the  foot  of  the  perpen- 
dicular, the  more  remote  is  the  greater. 

31.  Given  a  circle  and  a  line  perpendicular  to  its  plane  at  its 
center.    Prove  that  a  line  drawn  to  the  circle  from  any  point  of 
the  perpendicular  is  perpendicular  to  the  tangent  to  the  circle 
through  its  foot. 

HINT.    Draw  any  other  line  from  the  given  point  to  the  tangent  and 
apply  Exercise  30. 

Theorem  18 

426.  If  one  of  two  parallel  lines  is  perpendicular  to  a 
plane,  the  other  line  is  also  perpendicular  to  the  plane. 


Given  the  line  AB  parallel  to  the  line  KF  and  perpendicular 
to  the  plane  M. 

To  prove  that  KF  is  _L  to  M. 

Proof.  KF  intersects  M.  §  402 

Draw  FB,  and  any  other  line  in  M  through  F,  as  FG.    Draw 

BH  in  M  II  to  FG. 

AB  is  _L  to  BF.  Why? 

Hence  KF  is  J_  to  BF.  §  47 


BOOK  VI  337 
It  remains  to  prove  KF  _i_  to  FG. 

Now               AB  is  II  to  KF  and  EH  is  II  to  FG.  Why  ? 

Hence                           Z.ABH=^.KFG.  §407 

But                            ABH  is  a  right  angle.  Why? 

Therefore                       KF  is  _L  to  FG.  Why? 

Hence                             /iCF  is  _L  to  717.  Why  ? 

QUERY  1.  In  the  preceding  proof  why  is  it  not  sufficient  to  prove  AB 
and  KF  both  perpendicular  to  BF,  and  then  to  refer  to  §  44  ? 

427.  Corollary  1.    If  two  lines  are  perpendicular  to  the  same 
plane,  they  are  parallel. 

HINTS.  Given  a  and  b  JL  to  M.  Draw 
c  through  the  foot  of  b  and  II  to  #.  Then  c 
is  _L  to  M  (§  426).  Therefore  b  and  c  coin- 
cide (§  421). 

428.  Distance  to  a  plane.    The  dis- 
tance from  a  point  to  a  plane  is  the  length  of  the  perpendicular 
from  that  point  to  the  plane. 

By  §  422  the  distance  from  a  point  to  a  plane  is  the  shortest 
distance  from  that  point  to  the  plane. 

The  distance  between  two  parallel  planes  is  the  distance  from 
a  point  of  one  to  the  other. 

429.  Corollary  2.  The  distances  to  one  of  two  parallel  planes 
from  any  two  points  of  the  other  are  equal. 

QUERY  2.  What  is  the  locus  of  points  a  given  distance  from  a  given 
plane  ? 

QUERY  3.  What  is  the  locus  of  points  equidistant  from  two  given 
parallel  planes? 

QUERY  4.  How  would  you  locate  the  points  in  a  given  plane  which 
are  6  inches  from  another  given  plane  ?  In  what  kind  of  figure  would 
the  points  lie  ? 


338  SOLID  GEOMETRY 

REVIEW  EXERCISES 

32.  Two  points  on  opposite  sides  of  a  plane  and  equally  dis- 
tant from  it  determine  a  line-segment  which  is  bisected  by  the 
foot  of  the  line. 

33.  Two  points  on  the  same  side  of  a  plane  and  equally  dis- 
tant from  it  determine  a  line  that  is  parallel  to  the  plane. 

34.  A  plumb-line  6  feet  long  is  suspended  from  the  ceiling  of 
a  room  8  feet  high.    The  lowest  point  of  the  line  is  30  inches  from 
a  given  point  on  the  floor.    What  is  the  distance  from  this  given 
point  to  the  point  in  the  floor,  directly  under  the  plumb-line  ? 

35.  A  point  is  16  inches  from  a  given  plane.    What  is  the 
perimeter  of  the  circle  which  contains  the  points  of  the  plane 
which  are  20  inches  from  the  point. 

36.  A  line  parallel  to  a  plane  is  everywhere  equidistant  from 
the  plane. 

37.  Construct  a  line  perpendicular  to  a  given  pair  of  parallel 
lines,  which  shall  also  meet  a  given  third  line.    Is  there  any  case 
in  which  the  construction  is  impossible  ? 

38.  Every  line  perpendicular  to  a  line  which  is  given  perpen- 
dicular to  a  given  plane  is  parallel  to  the  plane. 

39.  Let  0  denote  the  center  of  an  equilateral  triangle  ABC 
whose  side  is  a.    At  0  a  line-segment  OP  is  drawn  perpendicular 
to  the  plane  of  the  triangle,  so  that  the  angle  APB  is  a  right 
angle.    How  long  is  OP  ? 

40.  A  point  is  10  inches  from  a  given  plane.    Lines  20  inches 
long  are  drawn  from  the  point  to  the  plane.    What  is  the  area  of 
the  circle  on  which  the  feet  of  these  lines  lie  ? 

41.  What  is  the  locus  of  points  in  space  equidistant  from  two 
parallel  lines  ?    Prove  your  statement  correct. 

42 .  What  condition  must  be  satisfied  by  two  lines  in  order  that 
it  may  be  possible  to  construct  a  plane  containing  one  of  them 
and  perpendicular  to  the  other  ?   Assuming  that  the  condition  is 
satisfied,  perform  the  construction. 


BOOK  VI 


339 


ANGLES  BETWEEN  PLANES 

430.  Dihedral  angle.    A  dihedral  angle  is  the  figure  formed 
by  two  planes  which  meet  each  other  (§  390). 

QUERY  1.    Can  you  hold  your  book  so  as  to  form  a  dihedral  angle? 
QUERY  2.   How  many  dihedral  angles  are  formed  by  the  walls,  ceiling, 
and  floor  of  a  square  room  ? 

431.  Parts  of  a  dihedral.    The  portions  of  the  planes  which 
form  a  dihedral  angle  are  called  the  faces 

of  the  angle. 

The  intersection  of  the  faces  of  a  dihedral 
angle  is  called  its  edge. 

A  dihedral  angle  may  he  designated  by  its 
edge  when  there  is  no  ambiguity.  Thus,  we  may 
designate  the  adjacent  figure  as  the  dihedral 
AB.  It  may  also  be  designated  as  K-AB-R. 

QUERY  3.  Can  two  dihedral  angles  have  one 
face  in  common  ?  Illustrate  with  your  book. 

QUERY  4.    Can  more  than  one  dihedral  angle  have  the  same  edge  ? 

432.  Plane  angle.    The  plane  angle  of  a  dihedral  angle  is 
formed  by  two  lines,  one  in  each  face,  each  perpendicular  to 
the  edge  at  the  same  point. 

In  the  above  figure  CDE  and  FGII  each  represent  a  plane  angle 
of  the  dihedral  AB. 

EXERCISE  43.  Any  two  plane  angles  of  the  same  dihedral 
angle  are  equal  to  each  other. 

QUERY  5.   Is  the  plane  of  ED  and  DC  _L  to  ABt 

QUERY  6.  Can  angles  other  than  plane  angles  be  formed  by  two 
lines,  one  in  each  face  of  the  dihedral  angle  and  intersecting  its  edge 
at  the  same  point?  Illustrate. 

QUERY  7.  Can  you  find  very  small  and  also  very  large  (nearly  180°) 
angles  in  illustrating  Q.uery  6  ? 

QUERY  8.  What  relations  do  the  planes  of  the  plane  angles  of  a 
dihedral  angle  bear  to  the  edge  of  the  dihedral? 


340 


SOLID  GEOMETRY 
Theorem  19 


433.  If  two  dihedral  angles  are  congruent,  their  plane 
angles  are  equal. 


Given  the  congruent  dihedrals  AB  and  CD  and  their  plane 
angles  GFE  and  KLH  respectively. 

To  prove  that  Z  GFE  =  Z  KLH. 

Proof.  Since  the  dihedrals  are  congruent,  we  may  consider 
them  as  different  positions  of  the  same  figure.  §  378 

Hence,  if  they  are  brought  into  coincidence  so  that  the  points  F 
and  L  coincide,  FE  will  coincide  with  LH  and  FG  with  LK.  §  41 

Therefore  A  EFG  and  HLK  coincide  and  are  equal. 


Theorem  20 

434.  Two  dihedral  angles  are  congruent  if  their  plane 
angles  are  equal. 

Given  (see  figure  for  Theorem  19)  the  dihedrals  AB  and  CD, 
having  equal  plane  angles  GFE  and  KLH. 

To  prove  that  the  dihedral  angles  AB  and  CD  are  congruent. 

Proof.    Bring  the  equal  plane  A  into  coincidence.  §  20 

Then  the  plane  of  GFE  will  coincide  with  that  of  KLH.     Why  ? 


BOOK  VI  341 

The  edge  AB  is  _L  to  the  plane  of  GFE  at  F,  and  the  edge  CD 

is  _L  to  the  plane  of  KLH  at  L.  §  415 

Then  the  edge  AB  coincides  with  the  edge  CD,  §  421 

and  the  faces  A  G  and  CK,  and  AE  and  CH,  coincide.     §  382 
Therefore  the  dihedrals  AB  and  CD  are  congruent.          Why  ? 

QUERY  1.  Following  the  analogy  of  the  definition  of  vertical  angles 
in  plane  geometry,  can  you  give  a  definition  of  vertical  dihedral  angles  ? 

QUERY  2.    Why  are  vertical  dihedrals  equal  ? 

QUERY  3.  If  two  parallel  planes  are  cut  by  a  third  plane,  how  would 
you  pass  a  plane  which  would  determine  the  plane  angles  of  all  eight 
dihedrals  of  the  figure  ? 

QUERY  4.  Can  you  state  several  theorems  giving  relations  between 
the  dihedral  angles  formed  when  two  parallel  planes  are  cut  by  a 
third  plane? 

435.  Perpendicular  planes.    Two  planes  are  perpendicular 
to  each  other  if  they  form  a  dihedral  angle  whose  plane  angle 
is  a  right  angle. 

436.  Right  dihedral.    A  dihedral  angle  whose  plane  angle 
is  a  right  angle  is  called  a  rigid  dihedral  angle. 

NOTE.  If  two  equal  dihedral  angles  are  placed  adjacent  to  each 
other,  doubling  of  the  dihedral  angle  also  doubles  the  plane  angle. 
In  general  two  plane  angles  are  in  the  same  proportion  as  their 
corresponding  dihedrals. 

EXERCISES 

44.  Following  the  analogy  of  the  corresponding  definitions  of 
plane  geometry,  define  the  following  kinds  of  dihedral  angles : 
acute,  obtuse,  adjacent,  supplementary,  complementary. 

45'.  Construct  a  dihedral  angle  having  a  plane  angle  of  (a)  90°, 
(b)  45°,  (c)  60°. 

46.  Construct  a  right  dihedral  with  a  given  line  as  edge  and  a 
given  plane  containing  that  line  as  face. 

47.  If  two  planes  cut  and  are  perpendicular  to  each  other,  the 
four  dihedrals  formed  are  all  right  dihedrals. 


342 


SOLID  GEOMETRY 
Theorem  21 


437.  If  a  line  is  perpendicular  to  a  plane,  every  plane 
containing  the  line  is  perpendicular  to  the  plane. 


Given  the  line  AB  perpendicular  to  the  plane  Af,  and  PQ  any 
plane  containing  AB. 

To  prove  that  PQ  is  ±  to  M. 

Proof.  In  M  draw  EC  J_  to  the  edge  BQ  at  B. 

Then  AB  is  _L  to  BQ  and  to  BC.  Why? 

But       ARC  is  the  plane  Z  of  the  dihedral A-BQ-C.        Why? 

Therefore  PQ  is  J_  to  M.  §  435 

438.   Corollary.   If  a  line  is  perpendicular  to  a  plane,  every 
plane  parallel  to  the  line  is  perpen- 
dicular to  the  plane. 

HINT.    Apply  §§  399,  420,  and  437. 

QUERY  1.  In  Theorem  21  would  it 
have  been  correct  to  draw  BC  in  M 
perpendicular  to  AB  and  to  £Q? 

QUERY  2.    If  a  line  is  parallel  to  one    ^ 
plane  and  at  the  same  time  perpendicular 
to  another,  what  relation  must  the  planes  bear  to  each  other?  Illustrate. 

QUERY  3.    In  the  figure  above  what  is  the  plane  angle  of  the  dihedral 
formed  by  PQ  and  the  plane  of  AB  and  B C? 

QUERY  4.    How  many  planes  are  there  which  contain  a  given  per- 
pendicular to  a  plane  ? 


BOOK  VI 
EXERCISES 


343 


48.  If  a  plane  is  perpendicular  to  the  intersection  of  two  planes, 
it  is  perpendicular  to  each  of  the  planes. 

49.  If  three  lines  are  perpendicular  to  each  other  at  a  common 
point,  what  is  the  relation  to  one  another  of  the  three  planes 
determined  by  the  three  pairs  of  lines  ?    Prove  your  statement. 


Theorem  22 

439.  If  two  planes  are  perpendicular  to  each  other, 
a  line  drawn  in  one  perpendicular  to  their  intersection 
is  perpendicular  to  the  other. 


Given  the  plane  PQ  perpendicular  to  the  plane  M,  and  the  line 
AB  in  PQ  perpendicular  to  the  intersection  BQ. 

To  prove  that  AB  is  J_  to  M. 

Proof.    Draw          BC  in  M  _L  to  BQ  at  B. 

Then         ABC  is  the  plane  Z  of  dihedral  A-BQ-C.         Why? 

Hence  ABC  is  a  right  Z.  §  436 

Therefore  ABis_LtoM.  Why? 

QUERY  1.  If  two  planes  are  perpendicular  to  each  other,  any  line 
perpendicular  to  one  of  them  is  how  related  to  the  other? 

QUERY  2.  What  condition  must  be  fulfilled  in  order  that  there  may 
be  a  plane  perpendicular  at  the  same  time  to  a  given  plane  and  to  a 
given  line? 


344  SOLID  GEOMETRY 

EXERCISES 

50.  If  a  line  and  a  plane  are  both  perpendicular  to  the  same 
plane,  they  are  parallel.    (Assume  that  the  line  does  not  lie  in 
the  first  plane.    See  Theorem  21.) 

51.  Construct  a  plane  which  contains  a  given  point,  is  parallel 
to  a  given  line,  and  is  perpendicular  to  a  given  plane. 

Theorem  23 

440.  If  two  planes  are  perpendicular  to  each  other, 
a  line  drawn  from  any  point  in  one,  perpendicular  to 
the  other,  lies  in  the  first. 


Given  the  plane  PQ  perpendicular  to  the  plane  Af,  and  the 
point  A  in  PQ,  from  which  a  line  AB  is  drawn  perpendicular  to  M. 

To  prove  that  AB  lies  in  PQ. 

Proof.    From  A  draw  AB'  in  PQ  J_  to  the  intersection  RT. 
Then  AB' is  ±  to  M.  §  439 

Then  AB  and  AB'  are  the  same  line.  §  421 

Hence  AB  lies  in  PQ,  since  it  coincides  with  ,4 B',  which  was 
constructed  in  PQ. 

EXERCISE  52.  Construct  a  figure  and  devise  a  proof  of  Theo- 
rem 23  for  the  case  where  the  point  A  is  in  the  intersection,  RT, 
of  the  plane  PQ  and  M. 


BOOK  VI 


345 


Theorem  24 


441.  If  two  intersecting  planes  are  perpendicular  to  a 
third  plane,  their  line  of  intersection  is  perpendicular  to 
that  plane.  p 

m 


R 


Given  the  planes  PQ  and  /?S,  each  perpendicular  to  M,  and 
AB,  their  intersection. 

To  prove  that  AB  is  J_  to  M. 

Proof.    From  A,  a  point  of  AB,  drop  a  _L  AK  to  M. 

Then  AK  lies  in  PQ  and  also  in  RS.  §  440 

Hence  it  is  their  intersection  and  coincides  with  AB.         §  391 
Consequently  AB  is  _L  to  M,  since  it  coincides  with  the  line  A  K, 

which  is  drawn  _L  to  M. 

QUERY.    A  plane  revolves  about  a  fixed  line  which  lies  in  it  (as  the 

lid  of  a  chest  revolves  about  the  line  of  its  hinges).   Find  a  plane  to 

which  the  revolving  plane  is  always  perpendicular.     How  many  such 

planes  are  there  ? 

442.  Bisector  of  a  dihedral  angle.  A  plane  is  said  to  bisect 
a  dihedral  angle  if  it  contains  the  edge  of  the  dihedral  angle, 
and  if  the  dihedral  angles  which  it  forms  with  the  faces 
are  equal. 

EXERCISES 

53.  Construct  a  plane  bisecting-  a  given  dihedral  angle. 

54.  If  one  plane  meets  another,  forming  two  adjacent  dihedral 
angles,  the  planes  which  bisect  these  angles  are  perpendicular  to 
each  other. 


346  SOLID  GEOMETRY 

Theorem  25 

443.  The  bisector  of  a  dihedral  angle  is  the  locus  of 
points  equidistant  from  the  faces  of  the  dihedral. 


Given  the  dihedral  M-QO-N,  and  the  plane  PQ  bisecting  this 
dihedral. 

To  prove  (1)  that  every  point  in  PQ  is  equidistant  from  M 

and  N  ; 

(2)  that  every  point  equidistant  from  M  and  N  lies 
in  PQ. 

Proof.   (1)  Let  A  be  any  point  in  PQ. 
Draw  AB  and  AD  _L  to  M  and  to  N  respectively. 
Pass  the  plane  determined  by  AB  and  AD,  cutting  717,  PQ,  and 
N  in  BC,  AC,  and  CD  respectively. 

The  plane  ABCD  is  _L  to  M  and  to  N.  §  437 

Hence  OC  is  _L  to  plane  ABCD.  §  441 

Therefore  OC  is  _L  to  CB,  CA,  and  CD.  Why  ? 

Therefore  A  CB  and  A  CD  are  the  plane  angles  of  their  respective 
.dihedrals.  §432 

Hence  ^ACB=Z.ACD.  §433 

Therefore          A  ABC  is  congruent  to  A  DC  A.  §  50 

Hence  AB=AD.  Why? 

(2)  Let  A  be  any  point  such  that  the  J§,4Z)  and  AB  are  equal. 
Pass  the  plane  PQ  determined  by  A  and  OQ. 


BOOK  VI 


347 


Also  pass  the  plane  determined  by  AB  and  AD,  cutting  A/,  PQ, 
and  N  in  BC,  A  C,  and  CD  respectively. 

AABC  is  congruent  to  AACD.  Why  ? 

Therefore  Z  .4  CD  =  Z  .1  C£.  Why  ? 

But  these  angles  may  be  proved  to  be  the  plane  angles  of  their 
respective  dihedrals  by  the  same  method  as  that  used  earlier  in 
this  demonstration. 

Hence    dihedrals  A-CQ-B  and  A-CQ-D  are  equal,  §  434 

and  PQ  bisects  M-QO-N.  §  442 

Hence  A  lies  in  the  bisecting  plane. 

Theorem  26 

444.  If  a  line  is  not  perpendicular  to  a  plane,  one  and 
only  one  plane  can  be  passed  containing  the  line  and 
perpendicular  to  the  plane. 


Given  the  line  AB,  not  perpendicular  to  the  plane  M. 

To  prove  that  one  and  only  one  plane  can  be  passed  containing 
AB  and  _L  to  M. 

Proof.    From  any  point  C  of  AB  draw  CD  _L  to  M,  and  pass  the 
plane  A  Q  determined  by  AB  and  CD. 

A  Q  is  ±  to  M.  §437 

Any  other  plane  through  AB  _L  to  M  would  contain  CD,     §  440 
and  hence  would  coincide  with  A  Q.  §  382 

Therefore  one  and  only  one  plane  can   be  passed,  containing 
AB  and  J_  to  M. 


348  SOLID  GEOMETRY 

QUERY  1.  If  two  planes  cut  each  other,  forming  four  dihedrals,  what 
is  the  locus  of  points  equidistant  from  the  faces  of  these  dihedrals  ? 

QUKRY  2.  How  can  you  find  a  point  in  a  given  line  equidistant 
from  the  faces  of  a  given  dihedral  ?  Discuss  any  special  cases. 

QUERY  3.  What  is  the  locus  of  points  in  a  given  plane  equidistant 
from  the  faces  of  a  dihedral  angle  ?  Discuss  any  special  cases. 

QUERY  4.  What  is  the  locus  of  points  equidistant  from  the  faces  of 
a  dihedral  and  also  equidistant  from  two  given  points  ? 

QUERY  5.  What  is  the  locus  of  points  equidistant  from  the  faces  of 
a  dihedral  angle  and  also  equidistant  from  three  given  points? 

QUERY  6.  How  would  you  find  the  locus  of  points  4  inches  from 
each  of  the  faces  of  a  given  dihedral  ? 

QUERY  7.  What  is  the  locus  of  points  equidistant  from  two  parallel 
planes  and  also  equidistant  from  the  faces  of  a  given  dihedral? 

EXERCISES 

55.  If  three  or  more  planes  intersect  in  a  common  line,  the 
lines  perpendicular  to  them  from  any  given  external  point  are 
coplanar. 

56.  If  from  any  point  within  a  dihedral  angle  lines  are  drawn 
perpendicular  to  the  faces,  the  angle  between  these  lines  is  the 
supplement  of  the  plane  angle  of  the  dihedral. 

57.  All  the  perpendiculars  to  a  plane  erected  from  points  of  a 
line  in  that  plane  lie  in  a  plane  perpendicular  to  the  given  plane. 

58.  Draw  a  figure  and  devise  a  proof   of   Theorem   26  for 
the  case  where  AB  lies  in  M. 

59.  A  line  is  perpendicular  to  the  bisector  of  a  dihedral  at  the 
point  C  and  intersects  the  faces  of  the  dihedral  at  A  and  B.   Show 
that  AB  is  bisected  at  C. 

60.  If  the  bisecting  planes  of  two  adjacent  dihedrals  are  per- 
pendicular to  each  other,  the  exterior  faces  of  the  adjacent  angles 
form  one  plane. 

61.  Through  a  point  O  of  the  edge  of  a  dihedral  angle  a  line 
OA  is  drawn  in  one  of  the  faces.    Construct  in  the  other  face  a 
line  OB  such  that  the  angle  A  OB  will  be  a  right  angle. 


BOOK  VI  349 

PROJECTIONS 

445.  Projection  of  a  point.    The  projection  of  a  point  on 
a  plane  is  the  foot  of  the  perpendicular  from  the  point  to 
the  plane. 

The  perpendicular  which  contains  the  point  and  its  pro- 
jection is  called  the  projecting  line  of  the  point. 

QUERY  1.  Does  a  given  point  have  more  than  one  projection  on  a 
given  plane  ? 

QUERY  2.  Under  what  conditions  may  two  points  have  the  same 
projection  on  a  given  plane? 

QUERY  3.  If  two  points  are  projected  on  the  same  plane,  what  is  the 
relation  between  their  projecting  lines  ? 

QUERY  4.  If  a  point  is  projected  on  two  planes,  how  must  the  planes 
be  situated  in  order  that  the  projecting  lines  may  be  identical  ? 

446.  Projection  of  a  line.    The  projection  of  a  line  (or  of 
a  curve)  on  a  given  plane  is  the  locus  of  the  projections 
of  its  points  on  that  plane. 

The  plane  containing  a  given  line, 
and  perpendicular  to  a  given  plane 
(§  444),  is  called  the  projecting  plane 
of  the  line  on  the  given  plane. 

P  is  the  projecting  plane  of  the 
line  a  on  the  plane  M. 

QUERY  5.  Can  a  complete  line  have  a  line-segment  for  its  projection? 
Illustrate. 

QUERY  6.  What  is  the  projection  on  a  plane  of  a  line  perpendicu- 
lar to  it? 

QUERY  7.  In  what  part  of  the  heavens  is  the  sun  if  the  shadow  of 
a  line  is  its  projection  on  the  ground? 

QUERY  8.  Between  what  limits  may  the  length  of  the  projection  of 
a  given  line-segment  vary  ?  Illustrate. 

QUERY  9.  Under  what  conditions  is  the  projection  of  a  circle  an 
equal  circle  ?  a  line-segment  ? 

QUERY  10.  Under  what  conditions  does  the  projection  of  a  plane  figure 
form  a  straight  line  ? 


350 


SOLID  GEOMETRY 


Theorem  27 

447.  If  a  line  is  not  perpendicular  to  a  given  plane,  its 
projection  on  that  plane  is  the  intersection  of  its  projecting 
plane  with  the  given  plane. 


Given  the  plane  M,  the  line  AB  not  perpendicular  to  M,  and 
the  projecting  plane  SP,  which  intersects  M  in  OP. 

To  prove  (1)  that  every  point  of  AB  lias  its  projection  in  OP; 
(2)  that  every  point  of  OP  is  the  projection  of  some 
point  of  AB. 

Proof.  (1)  Let  C  be  any  point  of  AB}  and  let  F  be  the  foot  of 
the  J_  from  C  to  M. 

F  is  the  projection  of  C  on  M. 

CF  lies  in  SP. 
Therefore  F  is  in  OP,  the  intersection  of  M  and  SP.- 

(2)  Let  G  be  any  point  of  OP.    Draw  GH  _L  to  M. 
Then  GH  lies  in  SP. 

Hence     GH  must  either  cut  A  B  or  be  II  to  it. 
If  they  were  II,        AB  would  be  _L  to  M, 
which  contradicts  the  hypothesis. 

Therefore  GH  cuts  AB  in  some  point  K. 
Hence  G  is  the  projection  of  A'. 

QUERY  1.  Which  of  the  two  parts  of  the  proof  given  in  §  447  would 
it  be  possible  to  carry  out  in  an  attempt  to  show  that  the  projection 
of  a  circle  whose  plane  is  perpendicular  to  the  given  plane  is  a 
complete  line? 


§445 
§440 
§390 

§440 
§426 

§445 


BOOK  VI  351 

QUERY  2.    Is  the  projection  of  a  corkscrew  on  a  plane  ever  straight? 

QUERY  3.  By  holding  a  right  angle  in  various  positions,  what  angles 
may  be  obtained  as  its  projection  on  a  given  plane  ? 

QUERY  4.    In  what  case  does  a  rectangle  project  into  a  rectangle  ? 

QUERY  5.  Is  the  following  statement  correct?  "A  plane  is  deter- 
mined by  a  line  and  its  projection  on  a  given  plane."  Illustrate. 

EXERCISES 

62.  Construct  the  projection  of  a  given  line-segment  upon  a 
given  plane. 

63.  If  a  line-segment  is  parallel  to  a  plane,  it  is  parallel  and 
equal  to  its  projection  on  the  plane. 

64.  The  projection  of  a  square  upon  a  plane  which  is  parallel 
to  that  of  the  square  is  a  congruent  square. 

Theorem  28 

448.  The  acute  angle  which  a  line  makes  with  its  pro- 
jection on  a  plane  is  the  least  angle  which  it  makes  with 
any  line  drawn  in  the  plane  through  its  foot. 

B 


Given  the  line  AB,  and  AC  its  projection  on  the  plane  M.    Let 
AK  be  any  other  line  in  M  through  the  point  A. 

To  prove  that  Z.BAC<Z. BAK. 

Proof.    From  L,  any  point  on  AJE>}  draw  the  projecting  line  LF, 
and  draw  LR,  making  AR  =  AF. 
In&LAF  &nd  LAR, 

AL  =AL  and  AF  =  AR.  Why? 

But  LF<LR.  §422 

Therefore  /.LA  F  <  Z  LAR.  §  151 


352 


SOLID  GEOMETRY 


449.  Angle  between  a  line  and  a  plane.    The  angle  which  a 
line  makes  with  a  plane  is  the  acute  angle  which  it  makes 
with  its  projection  on  that  plane. 

QUERY.    If  a  line  is  oblique  to  a  plane,  what  is  the  greatest  angle 
that  it  makes  with  any  line  drawn  through  its  foot  in  the  plane  ? 

EXERCISES 

65.  A  line  makes  equal  angles  with  two  parallel  planes. 

66.  Two  parallel  lines  make  equal  angles  with  any  plane. 

67.  The  projections  of  two  equal  and  parallel  line-segments  on 
a  plane  are  equal. 

68.  Equal  line-segments  from  a  given  external  point  to  a  given 
plane  are  equally  inclined  to  the  plane. 

69.  What  is  the  length  of  the  projection  of  a  line-segment 
6  inches   long  on  a  plane  to  which  it  is   inclined  at  an  angle 
of  (a)  45°?  (b)  60°? 

70.  Two  lines  intersect  at  an  angle  of  60°,  and  each  makes  an 
angle  -of  45°  with  a  plane  M.    Prove  that  the  projections  of  the 
two  lines  on  M  are  perpendicular  to  each  other. 

450.  Projection  of  areas.    In  section  287  it  was  shown  that  if 
a  line-segment  AB  makes  an  angle  x  with  the  line  AR,  then 
the  projection  of  AB  on  AR  is 

A  C  =  A  B  •  cos  x. 

If  two  planes  make  an  angle  x 
with  each  other,  an  inspection  of  the 
adjacent  figure  shows  that  if  A  C 
denotes  the  projection  of  AB  on  M 

AC  =AB  .  cosx. 

If  now  a  rectangle  whose  base  and 
altitude  are  denoted  by  a  and  b  respec- 
tively is  placed  so  that  the  base  a  is  the 
line  of  intersection  of  the  plane  of  the 
rectangle  with  a  plane  with  which  it 


BOOK  VI 


353 


makes  an  angle  x,  then  the  projection  of  the  rectangle  is  another 
rectangle  whose  base  is  a  and  whose  altitude  is  bcosx.  Hence 
the  area  of  the  projection  of  the  orig- 
inal rectangle  is  ab  cos  x ;  that  is, 

area  PR  =  ab  cos  x  =  area  PQ  -  cos  x. 

Since  any  plane  figure  can  be  divided, 
either  exactly  or  approximately,  into  rec- 
tangles, it  can  be  proved  that  the  area 
of  the  projection  of  any  plane  figure  on 
a  plane  making  an  angle  x  with  the 
plane  of  the  original  figure  equals  the  area  of  the  original  figure 
multiplied  by  the  cosine  of  the  angle  between  the  planes. 

If  the  plane  of  a  circle  0  of  radius  r  makes  an  angle  x  with 
another  plane,  the  projection  of  the  circle  on  that  plane  is  a  closed 
curve  E  which  is  shorter  one  way 
than  it  is  the  other.  This  curve  is 
called  an  ellipse.  Evidently  AB,  the 
longest  axis  of  the  ellipse,  equals 
the  diameter  of  the  circle,  while  the 
shortest  axis,  CD,  is  the  projection 
of  the  diameter  of  the  circle.  Thus 
AM  =  RL  =  r,  while  CM '=  r  •  cos  x. 
If  the  projection  of  the  radius  on 
P  is  called  b,  then 

area  of  ellipse  =  area  of  projection  of  circle  —  (area  of  circle)  •  cos  x 
=  Trr2  •  cos  x  =  Trr  (r  cos  x)  =  irrb. 

It  is  customary  to  denote  the  longer  (major)  axis  of  the  ellipse 
by  2  a  and  the  shorter  (minor)  axis  by  2  b.  Then  the  area  of  the 
ellipse  is  represented  by  vab. 


EXERCISES 

71.  The  roof  of  a  house  slants  at  an  angle  of  45°.  When  the 
sun  iff  directly  overhead,  what  area  on  the  floor  is  struck  by  the 
sun  through  a  window  in  the  roof  5  feet  by  7  feet  ? 


354  SOLID  GEOMETRY 

72.  What  angle  does  one  plane  make  with  another  if  the  figures 
of  one  of  them  are  projected  on  the  other  into  figures  of  half  the 
area  of  the  original  figures  ? 

73.  Find  the  area  of  an  ellipse  whose  axes  are  5  and  9  inches 
respectively. 

74.  A  circle  of  radius  10  inches  lies  in  a  plane  which  makes 
an  angle  of  60°  with  a  second  plane.    What  is  the  area  of  the 
projection  of  the  circle  on  the  second  plane  ? 

75.  If  the  projection  of  a  circle  of  diameter  24  inches  is  an 
ellipse  whose  axes  are  24  and  16  respectively,  find  the  angle  which 
the  plane  of  the  circle  makes  with  the  plane  of  the  ellipse. 

QUERY  1.  Keeping  in  mind  the  preceding  discussion  of  projections, 
what  reason  can  you  assign  for  the  fact  that  it  is  hotter  in  summer 
than  it  is  in  winter? 

SKEW  LINES 

451.  Skew   lines.    Two    nonintersecting    and    nonparallel 
lines  in  space  are  called  skew  lines. 

QUERY  2.  Among  the  lines  meeting  two  skew  lines  is  it  possible  to 
find  a  pair  which  (1)  intersect,  (2)  are  parallel? 

452.  Skew  quadrilateral.    The  figure  formed  by  the  line- 
segments  joining  four  noncoplanar  points  in  order  is  called 
a  skew  quadrilateral. 

If  a  quadrilateral  is  cut  from  a  sheet  of  paper  and  folded  along 
a  diagonal,  the  properties  of  the  skew  quadrilateral  may  easily  be 
visualized. 

QUERY  3.    Can  two  sides  of  a  skew  quadrilateral  be  parallel? 

QUERY  4.    Do  the  diagonals  of  a  skew  quadrilateral  intersect? 

QUERY  5.    Can  a  skew  quadrilateral  have  four  right  angles  ? 

QUERY  6.  Explain  how  a  skew  quadrilateral  may  have  four  equal 
sides,  two  opposite  right  angles,  and  the  other  two  angles  acute. 

QUERY  7.  What  is  the  locus  of  points  viewed  from  which  a  skew 
quadrilateral  looks  like  a  plane  angle  ? 


BOOK  VI  355 

Construction  S 
453.  To  construct  a  line  perpendicular  to  two  skew  lines. 


Given  the  skew  lines  AB  and  CD. 

Required  to  construct  a  line  perpendicular  to  loth  AB  and  CD. 

Construction.    Through  R,  any  point  of  CD,  draw  a  line  R S 
II  to  AB,  and  pass  the  plane  M  determined  by  CD  and  RS. 

The  plane  M  is  II  to  AB.  §  396 

Pass  a  plane  CK,  containing  CD  and  _L  to  M,  intersecting  AB 
at  the  point  P.  §  444 

In  CK,  from  the  point  P,  draw 

PQ  _L  to  CD.  §  235 

Then  PQ  is  also  _L  to  AB. 

Proof.    Through  PQ  and  ^45  pass  a  plane  meeting  AT  in  Q7\ 
Then  Q.T  is  II  to  AB.  §399 

But  PQ  is  _L  to  M.  §  439 

Hence  PQ  is  _L  to  QT.  Why  ? 

Therefore  PQ  is  _L  to  AB.  §  47 

454.  Corollary.  The  common  perpendicular  of  two  skew  lines 
is  the  shortest  line  that  connects  them. 

HINTS.  Let  PQ  be  the  common  _l_,  and 
let  KR  be  any  other  line  connecting  AB 
and  CD.  Let  M  contain  CD  and  be  II  to 
AB.  Draw  RT  _L  to  M.  Then  PQ  =  RT. 
Prove  RT<RK. 


356  SOLID  GEOMETRY 

REVIEW  EXERCISES 

76.  Construct  two  parallel  planes,  each  containing  one  of  two 
given  skew  lines. 

77.  If  a  plane  is  perpendicular  to  the  edge  of  a  dihedral  angle, 
it  is  perpendicular  to  each  of  the  faces. 

78.  If  one  of  two  planes  is  parallel  to  a  given  line,  but  the 
other  is  not,  the  planes  must  intersect. 

79.  If  three  equal  line-segments  which  do  not  all  lie  in  the 
same  plane  are  each  perpendicular  to  a  plane,  their  other  extremi- 
ties determine  a  plane  parallel  to  that  plane. 

80.  The  projections  of  two  parallel  lines  upon  a  plane  are  either 
two  parallel  lines  or  one  line  or  two  points. 

81.  If  one  side  of  a  right  angle  is  parallel  to  a  plane,  the  pro- 
jection of  the  right  angle  upon  the  plane  is  a  right  angle  or  a 
straight  line. 

82.  If  one  side  of  a  square  is  parallel  to  a  plane,  the  projection 
of  the  square  on  the  plane  is  a  rectangle  or  a  line-segment. 

83.  In  sawing  off  a  square  timber  with  a  handsaw  it  is  com- 
paratively easy  to  keep  the  blade  of  the  saw  perpendicular  to  one 
of  the  edges  of  the  timber.    If  this  is  done,  prove  that  the  stick 
is  sawed  off  square. 

84.  Given  a  room  12  feet  high,  what  is  the  locus  of  a  point  on 
a  15-foot  pole  5  feet  from  one  end  if  the  extremities  of  the  pole 
are  respectively  in  the  floor  and  the  ceiling  of  the  room  ? 

85.  If  two  parallel  lines  meet  two  parallel  planes,  the  four 
angles  which  the  lines  make  with  the  planes  are  equal. 

86.  If  two  planes  are  perpendicular  to  two  perpendicular  lines, 
each  to  each,  the  planes  form  a  right  dihedral  angle. 

87.  If  a  quadrilateral  has  four  right  angles,  it  lies  in  a  plane. 

88.  Through  a  given  point  construct  a  plane  making  equal 
angles  with  two  given  intersecting  planes. 


BOOK  VI  35T 

89.  If  two  lines  in  one  of  two  intersecting  planes  make  equal 
angles  with  the  intersection,  they  make  equal  angles  with  the 
other  plane. 

90.  The  sides  of  any  plane  angle  are  equally  inclined  to  any 
plane  through  its  bisector. 

91.  If  two  planes  are  not  perpendicular,  the  projection  on  one 
of  any  parallelogram  lying  in  the  other  is  a  parallelogram. 

92.  One  side  of  a  square  lies  in  a  plane  with  which  the  plane 
of  the  square  makes  an  angle  of  60°.    If  a  side  of  the  square  is 
8  inches,  what  is  the  area  of  its  projection  ? 

93.  One  side  of  an  equilateral  triangle  lies  in  a  plane  with 
which  the  plane  of  the  triangle  makes  an  angle  of  30°.    If  each 
side  of  the  triangle  is  10  inches,  what  is  the  area  of  its  projection  ? 


BOOK  VII 


POLYHEDRONS,  CONES,  AND  CYLINDERS 

455.  Polyhedron.    A  solid  bounded  by  polygons  is  called  a 
polyhedron. 

The  bounding  polygons  are  called  the 
faces,  their  lines  of  intersection  are  called 
the  edges,  and  the  intersections  of  the  edges 
are  called  the  vertices  of  the  polyhedron. 

456.  Convex  polyhedrons.  A  polyhedron 
which  lies  entirely  on  one  side  of  the 
planes  of  each  of  its  faces  is  called  a 
convex  polyhedron. 

Unless  the  contrary  is  stated,  it  will 
be  assumed  that  all  of  the  polyhedrons 
treated  in  this  book  are  convex. 


457.  Plane   sections.    The  intersection   of   a   solid  and  a 
plane  is  called  a  plane  section  of  the  solid.   The  plane  sections 
of  convex  polyhedrons  are  convex 

polygons.  A  plane  section  which 
has  a  circle  as  a  perimeter  is  called 
a  circular  section. 

458.  Prism.    A  prism  is  a  poly- 
hedron  two    of    whose    faces    are 

polygons  which  lie  in  parallel  planes,  and  whose  other  faces 
are  parallelograms  which  intersect  in  parallel  lines. 

358 


BOOK  VII 


359 


The  faces  which  lie  in  the  parallel  planes  are  called  the 
bases  of  the  prism.  The  parallelograms  included  between  the  bases 
are  called  the  lateral  faces.  The  intersec- 
tions of  the  lateral  faces  are  called  the 
lateral  edges.  The  sum  of  the  areas  of 
the  lateral  faces  is  called  the  lateral  area 
of  the  prism.  The  perpendicular  distance 
between  the  bases  is  called  the  altitude  of 
the  prism. 

In  the  adjacent  figure,  AC  and  FH 
are  the  bases,  AF,  BG,  etc.  are  the  lateral 
edges,  AG,  BH,  etc.  are  the  faces,  and  PQ  is  the  altitude. 

QUB:RY  1.  Why  is  each  side  of  the  upper  base  of  a  prism  necessarily 
parallel  to  a  side  of  the  lower  base  ? 

QUERY  2.    In  the  adjacent  figure  all  of  the  faces  are  parallelograms, 
and  the  faces  A  and  B  lie  in  parallel  planes. 
Why  is  it  not  a  prism  ? 

QUERY  3.  What  is  the  least  number  of 
faces  that  a  polyhedron  can  have?  edges? 
vertices  ? 

QUERY  4.  Is  there  any  greatest  number  of 
faces  that  a  polyhedron  can  have  ? 

QUERY  5.  If  the  lateral  edges  of  a  prism  make  a  very  small  angle 
with  a  base,  what  is  the  relation  between  the  length  of  the  altitude  and 
that  of  a  lateral  edge? 

QUERY  6.  Why  must  the  upper  and  the  lower  base  of  a  prism  have 
the  same  number  of  sides  ? 

QUERY  7.  If  two  prisms  have  equal  lateral  edges,  are  their  altitudes 
necessarily  equal? 

QUERY  8.    Can  a  plane  section  of  a  prism  be  a  trapezoid? 

EXERCISES 

1.  Any  lateral  face  of  a  prism  is  parallel  to  each  of  the  lateral 
edges  which  it  does  not  contain. 

2.  The  section  of  a  prism  made  by  a  plane  cutting  the  prism 
and  parallel  to  one  of  the  lateral  edges  is  a  parallelogram. 


SOLID  GEOMETRY 

Theorem  1 

459.  The  sections  of  a  prism  ly  two  parallel  planes, 
each  cutting  all  of  ike  lateral  edges,  are  congruent 
polygons. 


Given  any  prism  cut  by  parallel  planes  G-L  and  A-D,  form- 
ing the  sections  ABCDEF  and  GONLKH. 

To  prove  that  these  polygons  are  congruent,  that  is,  that  their 
corresponding  sides  and  angles  are  equal. 

Proof.              AB  is  II  to  GO,  B C  is  II  to  ON,  etc.  §  83 

Hence                      Z.ABC  =  Z  GON,  etc.  §  407 

Furthermore                  AB  =  GO,  EC  =  ON,  etc.  §  85 

Therefore      ABCDEF  is  congruent  to  GHKLNO.  §  24 

460.  Right  section.    If  a  plane  cutting  all  the  edges  of  a 
prism  is  perpendicular  to  one  of  them,  the  section  formed  is 
called  a  right  section. 

461.  Right   prism.    A  right  prism  is   one 
whose  base  is  a  right  section  of  the  prism. 

If  a  prism  is  not  a  right  prism,  it  is  said 
to  be  oblique. 

Prisms  whose  bases  are  triangles,  quadri- 
laterals, or  hexagons  are  called  triangular,  quadrangular,  or 
hexagonal  prisms. 


BOOK  VII 


361 


462.  Regular  prism.    A  right  prism  whose  base  is  a  regular 
polygon  is  called  a  regular  prism. 

Prove  each  of  the  following  properties  of 
a  prism : 

463.  The  lases  of  a  prism  are  congruent 
polygons. 

464.  All  sections  of  a  prism  parallel  to  the 
base  are  congruent. 

465.  The  lateral  edges  of  a  prism  are  equal  and  parallel. 

466.  A  right  section  of  a  prism  is  perpendicular  to  all  the 
lateral  edges. 

467.  The  lateral  faces  of  a  right  prism  are  rectangles. 

468.  The  altitude  of  a  right  prism  is  equal  to  a  lateral  edge. 

EXERCISES 

3.  If  a  right  section  of  a  prism  is  a  rectangle,  then  the  adja- 
cent lateral  faces  of  the  prism  are  perpendicular  to  each  other. 

4.  If  a  right  section  of  a  prism  is  an  equilateral  triangle,  then 
every  section  having  one  of  its  sides  parallel  to  one  of  the  sides  of 
the  given  section  is  an  isosceles  triangle. 

469.  Truncated  prism.    If  a  prism  is  cut  by  two  nonparallel 
planes  which  do  not  meet  inside  the  prism,  the  portion  of 
the   prism   between    the   planes  is    called  a 

truncated  prism. 

If  one  of  the  cutting  planes  is  perpen- 
dicular to  a  lateral  edge,  the  figure  is  called 
a  right  truncated  prism. 

QUERY  1.  What  kind  of  figures  are  the  lateral 
faces  of  a  truncated  prism  ? 

QUERY  2.    How  many  lateral  faces  of  a  truncated  prism  can  be 

parallelograms  ? 


362 


SOLID  GEOMETRY 


EXERCISE  5.  If  in  two  right  truncated  prisms  three  lateral 
edges  of  one  are  equal  respectively  to  three  lateral  edges  of  the 
other,  and  the  bases  to  which  these  edges  are  perpendicular  are 
congruent,  the  right  truncated  prisms  are  congruent. 

HINT.    Prove  by  superposition. 

470.  Parallelepiped.     A  prism  whose  bases  are  parallelo- 
grams is  called  a  parallelepiped. 

From  this  definition,  together  with  the  definition  of  a  prism, 
it  follows  that  all  the  faces  of  a  parallelepiped  are  parallelo- 
grams. It  follows  from  §  465  and  §  407  that  any  pair  of  opposite 
faces  of  a  parallelepiped  are  parallel.  Hence  we  may  consider  any 
two  opposite  faces  of  a  parallelepiped  as  its  bases. 

471.  Right  parallelepiped.     If  a  parallelepiped  is  a  right 
prism,  it  is  called  a  right  parallelepiped. 


PARALLELEPIPED 


RIGHT  PARAL- 
LELEPIPED 


RECTANGULAR 
SOLID 


CUBE 


We  shall  assume  that  the  base  of  a  right  parallelepiped  is  one 
of  the  two  faces  to  which  the  edges  are  perpendicular. 

472.  Rectangular  solid.   A  right  parallelepiped  whose  bases 
are  rectangles  is  called  a  rectangular  solid. 

The  lengths  of  the  edges  of  a  rectangular  solid  which  meet 
at  one  vertex  are  called  its  dimensions.  It  should  be  noted 
that  the  dimensions  of  a  rectangular  solid  are  numbers. 

473.  Cube.    If  three  edges   of  a  rectangular  solid   which 
meet  in  the  same  point  are  equal,  the  figure  is  called  a  cube. 

From  this  definition  it  appears  that  the  three  dimensions 
of  a  cube  are  equal  to  each  other. 


BOOK  VII  363 

474.  Diagonal,  A  diagonal  of  a  polyhedron  is  a  line-segment 
joining  two  vertices  which  do  not  lie  in  the  same  face. 

QUERY  1.  How  many  altitudes  does  a  parallelepiped  have  ?  Are  any 
two  of  them  necessarily  (1)  perpendicular,  (2)  equal  ? 

QUERY  2.   Are  any  two  altitudes  of  a  rectangular  solid  necessarily 

(1)  perpendicular,  (2)  equal  to  each  other? 

QUERY  3.    Are  each  of  the  lateral  edges  of  (1)  a  right  parallelepiped, 

(2)  a  rectangular  solid,  necessarily  equal  to  one  of  the  altitudes  ? 
QUERY  4.    Can  two  lateral  faces  of  (1)  a  parallelepiped,  (2)  a  rec- 
tangular solid,  be  equal  without  being  congruent  ? 

QUERY  5.  Are  the  adjacent  lateral  faces  of  (1)  a  rectangular  solid, 
(2)  a  right  parallelepiped,  perpendicular  to  each  other? 

QUERY  6.  How  many  diagonals  from  each  vertex  has  (1)  a  parallele- 
piped, (2)  a  prism  whose  base  is  a  pentagon  ? 

QUERY  7.  If  a  parallelepiped  has  a  rectangular 'base,  is  it  necessarily 
a  right  parallelepiped  ?  Is  it  necessarily  a  rectangular  solid  ? 

QUERY  8.   What  kind  of  figures  are  the  faces  of  a  cube?' 

QUERY  9.  Is  an  altitude  of  (1)  a  parallelepiped,  (2)  a  rectangular 
solid,  necessarily  equal  to  the  altitude  of  any  one  of  its  faces  ? 

QUERY  10.  Show  how  to  pass  a  plane  cutting  a  cube  so  that  the 
section  will  be  (1)  a  parallelogram,  (2)  a  square,  (3)  a  hexagon,  (4)  a 
triangle,  (5)  an  equilateral  triangle. 

EXERCISES 

6.  A  plane  which  contains  only  one  lateral  edge  of  a  prism  is 
parallel  to  all  of  the  other  edges. 

7.  Prove   that    if   two    cubes    have    equal    edges    they    are 
congruent. 

8.  Prove  that  the  three  diagonals  of  a  parallelepiped  meet  in 
a  point. 

9.  Prove  that  the  diagonals  of  a  rectangular  solid  are  equal  to 
each  other. 

10.  Prove  that  any  line  through  the  point  of  intersection  of 
the  diagonals  of  a  parallelepiped  and  terminated  by  opposite  faces 
of  the  parallelepiped  is  bisected  at  that  point, 


364 


SOLID  GEOMETRY 


11.  If  the  dimensions  of  a  rectangular  solid  are  a,  b,  and  c, 
what  is  the  length  of  the  diagonal  ? 

NOTE.  Strangely  enough  the  problem  of  this  exercise  is  not  solved 
in  any  of  the  works  of  the  ancient  geometers.  It  was  first  published 
in  1220  by  an  Italian  named  Fibonacci. 

12.  If  the  diagonal  of  a  cube  is  4  V3,  what  is  its  edge  ? 

13.  If  two  edges  of  a  rectangular  solid  are  6  and  8  respectively, 
and  the  diagonal  is  12,  what  is  the  third  edge  ? 

14.  Can  an  umbrella  38  inches  long  be  packed  in  a  trunk  whose 
inside  dimensions  are  34  x  24  x  12  inches  ? 

15.  If  the  angles  of  the  faces  meeting  at  a  given  vertex  of  a 
parallelepiped  are  80°,  120°,  and  50°  respectively,  what  are  the 
angles  at  the  other  vertices  ? 

16.  Find  the  sum  of  all  the  face  angles  of  a  parallelepiped. 

17.  If  the  altitude  of  a  parallelepiped  is  12  inches,  and  the 
lateral  edges  make  an  angle  of  30°  with  the  base,  find  the  length 
of  the  lateral  edges. 

18.  Show  that  the  diagonals  of  a  cube  do  not  meet  at  right 
angles. 

19.  Prove  that  the  mid-points  of  the  edges  of  a  cube,  which  are 
denoted  by  A,  B,  C,  D,  E,  F  in  the  adjacent 

figure,  are  all  equidistant  from  the  vertex  0 
and  are  the  vertices  of  a  regular  hexagon.    0 

20.  Prove  that  the  sum  of  the  plane 
angles  of  the  dihedrals  formed  by  the  faces 
of  a  parallelepiped  is  12  right  angles. 

21.  Prove  that  if  two  planes,  each  de- 
termined by  a  pair  of  lateral  edges  of  a 

prism,  meet,  their  intersection  is  parallel  to  the  lateral  edges. 

22.  Construct   a    parallelepiped  whose    edges  are    equal  and 
parallel  to  three  given  nonparallel  line-segments  in  space. 


BOOK  VII  365 

VOLUMES 

475.  Volume.  If  one  wishes  to  determine  with  a  moderate 
degree  of  accuracy  the  volume  of  an  open  vessel,  one  may 
fill  the  vessel  with  water  and  then  dip  the  water  out  with 
a  quart  measure.  The  number  of  times  the  quart  measure 
is  filled  denotes  the  volume  of  the  vessel  when  the  unit  of 
measure  is  one  quart.  This  number  might  be,  not  an  integer, 
but  an  integer  plus  a  proper  fraction.  If  a  measure  in  the 
form  of  a  cube  one  inch  on  an  edge  were  used  instead  of  a 
quart  measure,  the  volume  would  be  a  different  number, 
because  the  unit  of  measure  is  different.  If  a  smaller  unit  of 
measure  is  taken,  the  volume  of  the  vessel  would  be  likely  to 
be  obtained  with  greater  accuracy,  since  the  amount  left  over 
after  the  last  full  measure  would  probably  be  less  than  when 
a  large  measure  is  used.  It  is  clear  that  any  unit  of  measure 
is  contained  a  certain  number  of  times  in  any  solid,  whatever 
its  size  and  shape.  That  number  might  conceivably  be  an 
integer  or  a  fraction  or  even  an  irrational  number.  We  may 
thus  give  the  following  definition : 

The  volume  of  a  solid  is  the  number  of  times  it  contains  a 
given  solid  which  is  taken  arbitrarily  as  the  unit  of  volume. 

It  will  be  assumed  that  every  solid  which  is  considered  in 
this  text  has  a  volume.  One  object  of  the  study  of  solid 
geometry  is  to  find  an  expres- 
sion or  formula  for  the  volume 
of  some  of  the  simpler  solids. 

For  the  purpose  of  this  dis- 
cussion a  cube  whose  edge 
is  a  linear  unit,  as  one  inch 
or  one  centimeter,  will  be  taken  as  the  unit  of  volume. 

Consider  a  rectangular  solid  whose  edges  are  respectively 
3,  4,  and  2  inches.  Pass  planes  as  indicated  in  the  figure 


366  SOLID  GEOMETRY 

parallel  to  the  faces  of  the  solid  and  dividing  the  edges  into 
segments  one  inch  long.  These  planes  will  divide  the  base 
of  the  rectangular  solid  into  a  checkerboard  arrangement 
containing  3  x  4  =  12  squares,  each  of  which  is  the  base  of  a 
cube  formed  by  others  of  the  planes.  Since  the  altitude  of  the 
rectangular  solid  is  2,  there  will  be  2  layers  of  12  congruent 
cubes  each,  making  3  x  4  x  2  =  24  unit  cubes  in  all.  Hence 
the  unit  of  volume  is  contained  24  times  in  the  rectangular 
solid.  This  number  may  be  obtained  by  multiplying  together 
the  three  dimensions  of  the  rectangular  solid. 

If  a  rectangular  solid  has  edges  3J-,  4i,  21-  inches  respec- 
tively, it  would  be  more  convenient  to  take  as  a  unit  a  cube 
whose  edge  is  contained  evenly  in  each  of  the  given  edges. 
The  dimensions  of  this  rectangular  solid  may  be  written  ^-, 
_2_5_?  i_5_.  Taking  as  unit  a  cube  with  an  edge  |  of  an  inch  long, 
and  proceeding  as  before,  the  rectangular  solid  is  divided  into 
15  layers  of  little  congruent  cubes,  each  layer  containing 
20  x  25  of  these  unit  volumes.  Hence  the  volume  of  the 
cube  is  20  x  25  x  15  =  7500  when  the  unit  is  a  cube  of  edge 
l  inch.  If  it  is  desired  to  find  this  volume  in  cubic  inches, 
it  is  only  necessary  to  observe  that  there  are  6x6x6  =  216 
units  of  this  kind  in  a  cube  an  inch  on  a  side,  which  gives 
20x25x15  20  25  15 


ume  in  cubic  inches.    Hence  in  this  case  also  the  volume  of 
the  rectangular  solid  is  the  product  of  its  three  dimensions. 

When  the  three  dimensions  of  the  cube  are  numbers  which 
do  not  have  a  common  measure,  like  2,  v3,  "^6,  we  might 
get  the  approximate  volume  by  extracting  the  roots  indicated 
to  several  decimal  places,  and  taking  as  a  unit  a  cube  whose 
edge  is  a  common  measure  of  these  approximate  values  of 
the  edges.  The  approximate  volume  of  the  rectangular  solid 
which  is  found  is  a  little  too  small  or  too  great,  according  as 


BOOK  VII  367 

the  approximate  values  of  the  edges  are  taken  smaller  than 
or  greater  than  the  true  values ;  but  in  either  case  the  approxi- 
mate volume  of  the  rectangular  solid  is  found  by  taking  the 
product  of  the  approximate  values  of  its  edges.  This  leads  to 
the  following  assumption: 

476.  Assumption.     The  volume  of  a  rectangular  solid  is  the 
product  of  its  three  dimensions. 

Since  the  product  of  any  two  dimensions  of  a  rectangular 
solid  is  the  area  of  a  base,  we  may  state  the  preceding 
assumption  as  follows : 

The  volume  of  a  rectangular  solid  is  the  product  of  a  base  by 
the  corresponding  altitude. 

Prove  each  of  the  following  properties  of  rectangular  solids : 

477.  The  volumes  of  any  two  rectangular  solids  are  to  each 
other  as  the  products  of  their  three  dimensions. 

Proof.  Denote  by  V  and  F'  the  volumes  of  the  rectangular  solids 
whose  dimensions  are  a,  b,  c  and  a',  b',  c'  respectively.  Then 

F        abc 
V  =  abc  and  F'  =  a'b'c'.    Hence  -—.  = 


F'      a'o'c1 

478.  The  volumes  of  two  rectangular  solids  having  one  dimen- 
sion in  common  are  to  each  other  as  the  products  of  the  other  two. 

479.  TJie  volumes  of  two  rectangular  solids  having  two  dimen- 
sions in  common  are  to  each  other  as  their  third  dimensions. 

480.  Method  of  comparing  volumes.    Suppose  we  form  sev- 
eral piles  of  cards  2x3  inches,  each  pile  being  4  inches  high. 
The  piles  may  be  in  the  shape  either  of  a  rectangular  solid, 
or  of  a  parallelepiped,  or  of  some  other  irregular  solid.    But 
since  the  various  piles  consist  of  the  same  number  of  cards 
which  are  just  alike,  they  must  have  the  same  volume. 


308 


SOLID  GEOMETRY 


The  fact  that  all  of  the  cards  have  the  same  area  is  expressed 
in  geometric  language  by  saying  that  all  plane  sections  of 
each  of  the  figures  parallel  to  their  bases  are  equal  in  area. 


But  the  cards  need  not  all  be  equal  in  order  to  afford 
piles  of  equal  volume.    If  we  have  a  pile  of  cards  in  the  form 


of  a  pyramid,  we  may  distort  it  in  any  of  the  ways  indicated 
in  the  diagrams  without  affecting  the  volume  of  the  solid. 


Moreover,  the  sheets  need  not  be  of  the  same  shape.  A 
square  card  2  inches  on  a  side  has  the  same  volume  as  a 
circular  card  of  the  same  thickness  whose  area  is  4  square 


BOOK  VII  369 

inches  or  a  triangular  card  of  the  same  area.    Hence  a  pile  of 
100  of  the  square  cards  would  have  the  same  volume  as  a  pile 
of  an  equal  number  of  the  round  ones  or  of  the  triangular  ones. 
We  now  assume  without  proof  Cavalieri's  theorem: 

481.  Cavalieri's  theorem.  If  in  two  solids  of  equal  altitude 
the  sections  made  l>y  planes  parallel  to  and  at  the  same  dis- 
tance from  their  respective  bases  are  always  equal,  the  solids 
have  the  same  volume. 


It  should  be  emphasized  that  the  planes  mentioned  in  the 
theorem  may  be  any  distance  from  the  bases  of  the  two  figures,  but 
must  be  the  same  distance  from  the  base  of  each.  In  terms  of  the 
piles  of  cards,  this  means  that  cards  the  same  distance  up  in  each 
pile  must  have  the  same  size  in  order  that  the  piles  may  have 
equal  volumes. 

NOTE.  The  Italian  mathematician  Bonaventura  Cavalieri  was  born 
in  1598  and  died  in  1647  at  Bologna. 

The  principle  known  under  his  name  was  first  used  by  him  for  the 
case  of  plane  figures.  That  is  to  say,  he  stated  the  theorem  which  is 
given  in  the  preceding  section  in  such  a  way  as  to  include  not  only  the 
volume  of  certain  solids  but  also  the  area  of  certain  plane  figures.  The 
form  in  which  he  stated  his  theorem  is  as  follows : 

Plane  and  solid  figures  are  equal  in  content  when  sections  drawn  at 
the  same  height  from  the  base  produce  equal  lines  or  areas. 

A  rigorous  demonstration  of  the  validity  of  this  assumption  can  be 
given  by  use  of  the  methods  of  the  calculus. 


370 


SOLID  GEOMETEY 


QUERY  1.  If  two  prisms  have  equal  bases,  are  the  sections  of  the 
two  figures  made  by  planes  parallel  to  the  bases  necessarily  equal  ? 

QUERY  2.  How  manygparallelograms  are  there  with  a  given  base 
and  altitude?  AVhat  can  be  said  about  the  volumes  of  the  parallele- 
pipeds having  the  same  altitude,  which  are  constructed  on  these  paral- 
lelograms as  bases? 

QUERY  3.  If  a  pyramid  and  a  prism  have  congruent  bases  and  equal 
altitudes,  would  sections  the  same  distances  from  their  bases  be  con- 
gruent ?  Do  you  think  that  the  two  figures  have  the  same  volume  ? 

Theorem  2 

482.  Two  prisms  having  equal  bases  and  equal  altitudes 
are  equal  in  volume. 


Given  any  two  prisms  P  and  /?,  having  equal  altitudes  a, 
and  bases  B  and  C,  which  are  equal  in  area. 

To  prove  that    P  and  R  are  equal  in  volume. 

Proof.   Pass  planes  II  to  the  bases  of  the  prisms  P  and  R,  at 
any  distance,  k,  from  the  corresponding  bases. 

Call  the  areas  of  the  sections  thus  formed  S  and  T  respectively. 

Then  S  =  B  and  T  =  C.  §  459 

But                                          B=C,  Given 

Hence                                     S  =  T.  §32 

Therefore                               P  =  R.  §481 


BOOK  VII 


371 


483.  Corollary.    A  plane  which  passes  through  the  opposite 
parallel  edges   of  a  parallelepiped   divides   it   into    two   equal 
triangular  prisms. 

EXERCISE  23.  Two  right  prisms  are  equal  if  three  faces  which 
meet  at  a  vertex  of  one  are  respectively  equal  to  three  faces 
similarly  placed,  which  meet  at  a  vertex  of  the  other. 

HINT.  The  faces  which  are  given  equal  must  include  among  them  a 
base  of  each  prism. 

Theorem  3 

484.  The  volume  of  any  prism  is  equal  to  the  product 
of  its  base  and  altitude. 


7 

1 

i  * 
i____4___ 

B     * 


Given  any  prism  P,  with  base  B  and  altitude  a. 
To  prove  that  volume  P  =  B  •  a. 

Proof.    Let  K  be  a  rectangular  solid  having  its  base  equal  to  B 
and  its  altitude  equal  to  a. 

Volume  K =  B-  a.  §476 

But  the  base  and  altitude  of  P  equal  those  of  K.  Const. 

Hence  volume  P  =  volume  A'.  §  482 

Therefore  volume  P  =  B  -  a.  Why  ? 

485.   Corollary.     The   volume  of  a  parallelepiped  equals  the 
product  of  the  area  of  any  face  and  the  altitude  on  that  face. 


372  SOLID  GEOMETKY 

QUERY  1.  Is  the  volume  of  any  parallelepiped  equal  to  the  product 
of  its  three  altitudes  ?  Give  an  example. 

QUERY  2.  In  what  case  is  the  volume  of  a  parallelepiped  equal  to 
the  product  of  three  of  its  edges  ? 

QUERY  3.  In  what  kind  of  a  triangular  prism  would  the  volume  be 
equal  to  one  half  the  product  of  its  three  edges? 

EXERCISES 

24.  Construct  a  rectangular  solid  whose  base  is  equal  in  area 
to  that  of  a  given  triangular  prism,  and  which  has  an  altitude 
equal  to  that  of  the  prism.    Give  a  reason  for  each  step. 

25.  Prove  that  the  volume  of  a  triangular  prism  is  equal  to  one- 
half  the  product  of  the  area  of  a  given  lateral  face  and  the  distance 
to  that  face  from  the  opposite  lateral  edge. 

Theorem  4 

486.  The  lateral  area  of  a  prism  equals  the  product  of 
a  lateral  edge  and  the  perimeter  of  a  right  section. 

X 


.     B 

Given  any  prism  A-X,  of  which  HLMOP  is  a  right  section, 
and  one  of  whose  lateral  edges  is  AR. 

To  prove  that  lateral  area  of  A~X  =  perimeter  of  H~0  >  AR. 

Proof.    Plane  H-0  is  _L  to  .47?,  BS,  CU,  etc.  §  466 

AR  is  _L  to  HL,  BS  to  LM,  CU  to  MO,  etc.          Why  ? 


BOOK  VII  373 

Hence  the  lateral  faces  of  the  prism  are  parallelograms,  each 
having  one  of  the  line-segments  HL,  LM,  MO,  etc.  as  its  altitude, 
and  each  having  a  base  equal  to  AR.  §  465 

Hence  area  AY=AR  .  PH, 

area,BR=BS.HL,  Why? 

areaCS  =  CU  •  ML. 


Adding,  area  (/I  Y+BR+CS-\ )=AR(PH+HL+LM-{ ), 

or  lateral  area  of  A -X  =  perimeter  of  H-0  x  AR.     §  458 

487.  Corollary.  The  lateral  area  of  a  right  prism  is  equal  to 
the  product  of  its  altitude  and  the  perimeter  of  its  base. 

QUERY.  If  the  shape  of  a  prism  is  varied  by  moving  the  upper  base 
about  in  its  plane  while  the  lower  base  remains  fixed  in  position,  does 
the  volume  of  the  prism  vary  ?  Does  the  lateral  area  vary  ? 

EXERCISES 

In  the  following  exercises,  h  represents  the  altitude,  n  the  num- 
ber of  sides  of  the  base,  b  one  side  of  the  base,  B  the  area  of 
the  base,  V  the  volume,  S  the  lateral  area,  and  T  the  total  area, 

of  a  prism. 

26.  Given  V=  50,  B=  25.    Find  h. 

27.  Given  h  =  8j,  B=6.    Find  V. 
In  Exercises  28-35  the  prism  is  regular. 

28.  Given  h  =  12,  b  =  3,  n  =  3.    Find  S. 

29.  Given  h  =  10,  b  =  4,  n  =  4.    Find  V. 

30.  Given  h  —  6f ,  S=  80,  n  =  4.    Find  V. 

31.  Given  V=  56,  5  =  14.    Find  h. 

32.  Given  V=  300,  n  =  3,  h  =  20.    Find  b. 

33.  Given  n  =  4,  h  =  6,  7=  1536.    Find  I. 

34.  Given  n  =  3,b  =  6,k  =  8.    Find  T. 

35.  Given  n  =  6,  b  =  8,  V=  128.    Find  T. 


374  SOLID  GEOMETKY 

36.  Find  the  lateral  area  of  a  right  prism  whose  base  is  a  rec- 
tangle 6x8  inches  and  whose  altitude  is  1  foot.    Find  also  the 
total  area. 

37.  Find  the  lateral  area  of  a  prism  whose  base  is  an  equilateral 
triangle  4  inches  on  a  side,  and  whose  altitude  is  10  inches.    Find 
also  the  total  area. 

38.  The  total  area  of  a  cube  is  216  square  inches.    Find  the 
edge  and  the  volume. 

39.  Sand  lies  against  a  vertical  wall,  reaching  a  point  3  feet 
high.    If  the  sand  just  lies  at  rest  with  its  surface  at  an  angle 
of  30°  with  the  horizontal,  how  many  cubic  feet  of  sand  are  there 
in  a  pile  20  feet  long  ?  Assume  that  the  ends  of  the  pile  are 
perpendicular  to  its  length. 

40.  What  is  the  weight  of  a  block  of  ice  24  x  24  x  18  inches 
if  ice  weighs  92  per  cent  as  much  as  water  ?   (Cubic  foot  of  water 
weighs  62.5  pounds.) 

41.  A  railway  cut  25  feet  deep  is  to  be  made  with  one  side 
vertical  and  the  other  inclined  at  an  angle  of  45°  to  the  vertical. 
The  bottom  is  to  be  30  feet  wide.    How  many  cubic  feet  of  mate- 
rial must  be  removed  per  running  foot  of  track  ? 

42.  Express  T  in  terms  of  V  for  a  cube. 

43.  If  d  is  the  diagonal  of  a  cube,  express  V  in  terms  of  d. 

44.  Express  V  in  terms  of  b  and  h  for  a  triangular  prism  whose 
base  is  equilateral. 

45.  Find  correct  to  two  decimal  places  the  edge  of  a  cube  equal 
in  volume  to  a  prism  whose  base  is  a  regular  hexagon  4  inches  on 
a  side  and  whose  altitude  is  equal  to  an  edge  of  the  cube. 

46.  The  altitude  and  base  of  a  parallelepiped  are  6  and  8  inches 
respectively,  and  a  lateral  edge  whose  length  is  10  inches  makes 
an  angle  of  60°  with  the  base.    Find  the  volume. 

47.  A  regular  hexagonal  prism  whose  lateral  edge  makes  an 
angle  of  60°  with  the  plane  of  its  base  has  an  altitude  of  12  feet, 
while  one  edge  of  the  base  is  4  feet.   What  is  its  lateral  area  ? 


BOOK  VII 


375 


CYLINDERS 

488.  Cylindrical  surface.  If  a  line  moves,  always  remain- 
ing parallel  to  its  first  position  and  always  cutting  a  fixed 
plane  curve  not  situated  in  the  plane  of  the  line,  the  surface 
generated  is  called  a  cylindrical  surface. 


The  moving  line  is  called  the  gener- 
ator ;  in  one  of  its  positions  it  is  called 
an  element ;  the  fixed  curve  is  called  the 
directrix  of  the  surface. 

From  the  definitions  just  given  it 
follows  that  the  elements  of  a  cylinder 
are  parallel  (§  406). 

In  this  text  the  directrix  will  be  assumed  to  be  a  closed  curve, 
although  in  more  advanced  mathematical  work  the  directrix  is 
often  an  open  curve,  or  even  a  curve  consisting  of  several  branches. 

A  convex  curve  is  one  which  a  straight  line  can  cut  in  no  more 
than  two  points.  A  convex  cylindrical  surface  has  a  convex  curve  as 
directrix.  Only  convex  cylindrical  surfaces  are  studied  in  this  text. 

489.  Corollary.    Each  point  of  a  cylindrical  surface  lies  on 
one  and  only  one  element  of  the  surface. 

HINT.    Suppose  a  certain  point  were  contained  by  two  elements. 

490.  Cylinder.    The    solid   bounded  by  a 
cylindrical  surface  and  two  parallel  planes 
cutting  the  elements  is  called  a  cylinder. 

The  terms  altitude,  base,  right  section,  lat- 
eral area,  volume,  and  right  cylinder  are  defined 
similarly  to  the  corresponding  terms  applied 
to  the  prism  (§  458). 

491.  Circular  cylinder.    If  a  cylinder  has  a  circular  right 
section,  it  is  called  a  circular  cylinder. 

EXERCISE  48.  Define  the  terms  mentioned  in  §  490. 


376  SOLID  GEOMETRY 

Theorem  5 

492.   The  sections  of  a  cylindrical  surface  by  two  paral- 
lel planes •,  one  of  which  cuts  an  element,  are  congruent. 


Given  any  cylindrical  surface  AB,  of  which  one  element  AT 
is  cut  by  the  plane  M,  which  is  parallel  to  the  plane  N. 

To  prove  that  the  sections  formed  are  congruent. 

Proof.    The  plane  M  cuts  all  the  elements  of  A  B.  §  402 

Take  any  three  points  P,  Ry  S  at  random  in  the  intersection 
of  the  surface  by  M,  and  let  PA',  R  Y,  SZ  be  the  elements  of  the 
surface  containing  these  points.  §  489 

Let  the  elements  PX,  RY,  and  SZ  cut  .ZV  in  the  points  /,  K, 
and  L  respectively.  §  403 

Let  the  intersections  of  M  and  N  with  the  planes  determined 
by  the  parallel  lines  PJ,  RK,  SL  be  PS,  JL,  etc. 

PS  is  II  to  JL.  §  393 

Hence           PL  is  a  parallelogram,  and  PS  =  JL.  §§  83,  85 
Similarly,                SR  =  LK  and  RP  =  KJ. 

Therefore          APRS  is  congruent  to  AJKL.  Why? 

Since  P,  R,  and  S  were  taken  at  random,  it  follows  that  if  the 
upper  section  is  applied  to  the  lower  one  so  that  PS  coincides 
with  JL,  any  other  point,  as  R,  of  the  upper  section  will  coincide 
with  the  point  of  the  lower  section  which  is  on  the  same  element. 


BOOK  VII 


37T 


Hence  any  point  of  the  upper  section  coincides  with  some  point 
.of  the  lower. 

Similarly  it  can  be  shown  that  any  point  of  the  lower  section 
coincides  with  some  point  of  the  upper. 

Therefore  the  upper  and  lower  sections  are  congruent. 

493.  Corollary.    The  bases  of  a  cylinder  are  congruent. 

Theorem  6 

494.  The  section  of  a  cylinder  made  ~by  a  plane  which 
contains  an  element  of  the  cylinder  and  a  point  of  the 
cylindrical  surface  not  in  this  element  is  a  parallelogram. 


Given  any  cylinder  C  and  a  plane  M  which  contains  an  ele- 
ment AB  and  a  point  O  of  the  cylindrical  surface  not  on  AB. 

To  prove  that  the  section  formed  is  a  parallelogram. 
Proof.    Denote  by  FH  the  element  through  0. 
Now  FH  is  II  to  AB. 

If  FH  cut  M,  AB  would  also  cut  M. 

Therefore  FH  lies  in  M  and  also  in  the  surface  'of  C. 

Let  M  intersect  the  bases  of  C  in  H B  and  FA. 

HB  is  II  to  FA. 

Hence  ABHFis  a  parallelogram  each  side  of  which  lies  in  both 
the  plane  and  the  surface  of  the  cylinder. 

Therefore  the  section  of  C  by  M  is  a  parallelogram. 


§488 
§402 

Why? 


378  SOLID  GEOMETRY 

QUERY  1.  Would  the  section  of  the  cylinder  made  by  the  plane 
determined  by  an  element  and  any  point  in  one  of  the  bases  be  a 
parallelogram  ? 

QUERY  2.  What  kind  of  cylinder  may  have  a  square  as  a  section? 
a  rectangle  ?  a  rhombus  ? 

EXERCISES 

49.  Every  section  of  a  right  cylinder  made  by  a  cutting  plane 
perpendicular  to  the  base  is  a  rectangle. 

50.  The  intersection  of  a  right  cylinder  and  a  plane  which 
passes  through  an  element  is  a  rectangle. 

495.  Inscribed  prism.    If  the  bases  of  a  prism  are  inscribed 
in  the  bases  of  a  cylinder,  and  the  lateral  edges  of  the  prism 
are  elements  of  the  cylinder,  the  prism  is  said 

to  be  inscribed  in  the  cylinder. 

QUERY  1.  Can  a  regular  prism  be  inscribed  in  any 
given  right  circular  cylinder? 

QUERY  2.  Can  a  regular  prism  be  inscribed  in  any 
given  circular  cylinder  ? 

QUERY  3.  Can  a  rectangular  solid  be  inscribed  in 
any  given  right  circular  cylinder? 

EXERCISES 

51.  Construct  a  triangular  prism  inscribed  in  a  given  right 
circular  cylinder,  giving  a  reason  for  each  step. 

52.  Construct  a  regular  hexagonal  prism  inscribed  in  a  given 
right  circular  cylinder,  giving  a  reason  for  each  step. 

496.  Volume  of  cylinder.    If  the  base  of  a  prism  which  is 
inscribed  in  a  cylinder  has  a  very  large  number  of  sides,  and 
each  side  is  very  short,  then  the  area  and  the  perimeter  of 
the  base  of  the  prism  are  approximately  equal  to  the  area 
and  the  perimeter  respectively  of  the  base  of  the  cylinder. 
By  taking  the  sides  sufficiently  short,  as  close  approximations 
as  may  be  desired  can  be  obtained. 


BOOK  VII 


379 


The  inscribed  prism  whose  base  is  almost  the  same  in  area 
as  that  of  the  circumscribing  cylinder  will  have  a  large  num- 
ber of  very  narrow  parallelograms  -as  its  lateral  faces,  and  its 
lateral  area  and  volume  will  differ  very  little  from  the  lateral 
area  and  volume  respectively  of  the  cylinder. 
But  the  volume  of  the  prism,  however  many 
lateral  faces  it  may  have,  is  the  product  of 
its  base  and  its  altitude,  and  the  lateral  area 
of  the  prism  is  the  product  of  the  perimeter 
of  a  right  section  and  a  lateral  edge.  Since 
the  volume  and  the  base  of  the  cylinder 
become  very  nearly  equal  to  the  volume  and 
the  base  respectively  of  the  inscribed  prism, 
when  the  number  of  its  faces  is  sufficiently 
increased,  and  since  the  altitudes  of  the  two  figures  are  iden- 
tical, we  are  led  to  the  following  statements,  which  we  here 
assume  without  proof,  but  which  can  be  demonstrated  by  the 
theory  of  limits. 

Theorem  7 

497.  The  volume  of  a  cylinder  equals  the  product  of  the 
area  of  its  base  and  its  altitude. 

498.  Corollary.    The  volume  of  a  right  circular  cylinder  is  equal 
to  7rr2h,  where  r  denotes  the  radius  of  the  base  and  h  the  altitude 
of  the  cylinder. 

Theorem  8 

499.  The  lateral  area  of  a  cylinder  is  equal  to  the  prod- 
uct of  the  perimeter  of  a  right  section  and  an  element. 

500.  Corollary  1.    The  lateral  area  of  a  right  circular  cylinder 
is  equal  to  2  Trrh,  where  r  denotes  the  radius  of  the  base  and  h 
the  altitude  of  the  cylinder. 


380  SOLID  GEOMETRY 

501.   Corollary  2.    The  total  area  of  a  right  circular  cylinder 

is  equal  to  on       ?,o      20       ^7    ,     -\ 

$=2  Trrh  +  2  TTT  =  2  ?rr (7i  -f  r). 

NOTE.  Many  theorems  concerning  the  prism  are  equally  true  when 
applied  to  the  cylinder.  Theorems  7  and  8  illustrate  this  important 
fact,  which  may  be  stated  in  general  terms  as  follows :  'Any  theorem 
regarding  the  prism  which  does  not  depend  on  the  number  of  lateral 
faces  is  true  for  the  cylinder. 

QUERY  1.  The  lateral  surface  of  a  cylinder  is  cut  along  an  element 
and  rolled  out  flat.  What  kind  of  figure  is  obtained  ? 

QUERY  2.  What  is  the  locus  of  points  a  given  distance  from  a 
given  line? 

QUERY  3.  What  is  the  locus  of  points  a  given  distance  from  two 
given  parallel  lines? 

QUERY  4.  What  is  the  locus  of  points  a  given  distance  from  a  given 
cylinder  ? 

QUERY  5.  What  is  the  locus  of  points  a  given  distance  from  a 
given  line  and  equidistant  from  two  parallel  planes  to  which  the  given 
line  is  perpendicular?  What  if  the  planes  are  not  perpendicular  to  the 
given  line? 

EXERCISES 

In  the  following  exercises,  all  of  which  relate  to  right  circular 
cylinders,  //  represents  the  altitude,  r  the  radius  of  the  base,  S  the 
lateral  area,  T  the  total  area,  and  V  the  volume  : 

53.  Given  r  =  2,  h  =  2.    Find  T. 

54.  Given  r  =  4,  h  =  9.    Find  T. 

55.  Given  r  =  5,  h  =  21.    Find  S. 

56.  Given  r  =  4,  F=42.    Find  S. 

57.  Given  T=  235,  r  =  3.    Find  V. 

58.  Given  S  =  121,  r  =  2.    Find  h. 

59.  Given  r  =  2,  V=  64.    Find  h. 

60.  Given  V=  T,  r=±7.    Find  S. 

61.  Given  S  =  V,  h=r.    Find  r. 


BOOK  VII 


381 


Theorem  9 

502.  If  a  rectangle  is  revolved  about  one  side  as  an  axis, 
the  figure  formed  is  a  right  circular  cylinder. 


Given  any  rectangle  ABKL  which  revolves  about  AB  as  an  axis. 
To  prove  that  the  figure  generated  is  a  right  circular  cylinder. 


Proof.     AL  and  BK  each  generate  a  plane  _L  to^lZ?. 
Therefore       the  planes  generated  are  parallel. 
The  points  K  and  L  describe  circles  in  these  planes. 
KL  generates  a  cylindrical  surface. 


§418 
Why  ? 
Why  ? 
§488 
Hence  the  figure  0-L  is  a  right  circular  cylinder.    Why  ? 

503.  Cylinder  of  revolution.     A  right  circular  cylinder  is 
often   called   a  cylinder  of  revolution  when  it  is  desired  to 
emphasize  the  foregoing  method  of  generation. 

504.  Axis  of  cylinder.    The  line  joining  the  centers  of  the 
bases  of  a  right  circular  cylinder  is  called  the  axis  of  the 

cylinder. 

EXERCISES 

62.  The  axis  of  a  cylinder  of  revolution  is  equal  and  parallel 
to  the  elements  of  the  cylinder. 

63.  The  axis  of  a  cylinder  of  revolution  passes  through  the 
center  of  every  plane  section  of  the  cylinder  parallel  to  its  base. 

505.  Similar  cylinders.    If  two  cylinders  of  revolution  are 
generated  by  the  rotation  of  similar  rectangles  about  corre- 
sponding sides,  the  cylinders  are  said  to  be  similar. 


382 


SOLID  GEOMETRY 


Theorem  10 


506.  If  two  cylinders  of  revolution  are  similar, 

(a)  their  volumes  are  proportional  to  the  cubes  of  their 
altitudes  or  of  their  radii ; 

(b)  their  lateral  surfaces,  or  their  total  surfaces,  are  pro- 
portional to  the  squares  of  their  altitudes  or  of  their  radii. 


Given  any  two  similar  cylinders  of  revolution  with  radii 
r  and  rx,  and  altitudes  h  and  hlt  respectively. 

Let  V  and  Vv  S  and  Sv  T  and  771  represent  the  volumes,  lateral 
surfaces,  and  total  surfaces,  respectively. 

3         8 
To  prove  «- 


i  r  ) 

(»)     === 


Proof. 


Trr-h 


But 

Hence,  from  (1)  and  (2), 
V        r2 


(1)  §  498 

(2)  §§505,268 


(*)  T  = 


r! 

2  7rrh        r     h 


hi 

h2 


=  7-f  =  -4  =  71          (3)     §500 


and 


T_        27rr(r  +  h)     _  r_    ^_ 


7T1\  (l\  +  7tl) 


'I  +   />! 


501 


BOOK  VII 


383 


But  from  (2), 


r  +  h 

r*l  -f-  AJ 


(5)     §§257,262 


Hence,  from  (4),  (5),  and  (2), 

T_  _r_      r  +  h 

T'  ~  r,     r ,  -\-  h, 


507.  Theorem  of  Pappus.  If  a  circle  is  revolved  about  a  line 
lying  in  its  plane  but  not  intersecting  it,  a  figure  called  an  anchor 
ring  or  torus  is  formed.  If  one  imagines  the  torus  cut  along  one 
of  the  generating  circles  and  straightened  out  so  as  to  form  a 


cylinder,  one  would  expect  that  the  altitude  of  the  cylinder  would 
be  the  line  described  by  the  center  of  the  generating  circle  in  its 
revolution.  That  this  is  the  case  can  be  proved  by  means  of  the 
calculus;  in  fact,  a  much  more  general  theorem  can  be  demon- 
strated, which  is  called 

The  Theorem  of  Pappus.  If  a  closed  curve  (or  polygon)  rotates 
about  an  axis  lying  in  its  plane,  the  volume  of  the  ring  described 
equals  that  of  the  cylinder  (or  prism)  whose  base  is  the  generating 
figure  and  whose  altitude  is  the  length  of  the  curve  described  by  the 
center  of  gravity  of  the  generating  figure. 

The  area  of  the  surface  of  the  ring  generated  equals  the  lateral 
area  of  the  cylinder  (or  prism)  described. 


384  SOLID  GEOMETRY 

NOTE.  This  theorem  was  first  discovered  by  Pappus  of  Alexandria 
(third  century  of  the  Christian  era).  His  work  was  forgotten  for  more 
than  twelve  hundred  years,  until  the  interest  in  the  subject  was  revived 
at  the  end  of  the  sixteenth  century  by  the  works  of  Kepler  and  Guldin. 

Kepler  (1571-1630)  investigated  a  number  of  solids  generated  by  the 
rotation  of  a  plane  figure  and  succeeded  in  finding  rules  for  computing 
their  volumes  in  certain  particular  cases.  All  of  his  rules  are  special 
cases  of  the  Theorem  of  Pappus,  although  Kepler  never  announced  the 
theorem  in  its  general  form. 

Among  the  solids  treated  by  Kepler  were  the  torus  and  the  solid 
termed  by  him  w  the  apple,"  which  is  formed  by  revolving  a  segment 
greater  than  half  a  circle  around  its  chord  as  an  axis,  and  "  the  lemon," 
which  is  formed  by  revolving  a  segment  of  a  circle  with  an  arc  less  than 
180°  around  its  chord  as  an  axis.  The  Jesuit,  Paul  Guldin  (1577-1643), 
who  was  a  professor  of  mathematics  in  Rome  and  in  Graz,  was  the  first 
after  Pappus  to  give  a  statement  of  the  theorem  in  its  general  form  in 
a  work  published  in  1640.  He  failed,  however,  to  give  a  satisfactory 
proof  of  the  general  case. 

The  Theorem  of  Pappus  was  later  generalized  by  Leibniz  and  Euler. 
Leibniz  (1646-1716)  noticed  that  the  theorem  holds  true  when  the 
moving  plane  figure  moves  along  any  path  to  which  it  always  remains 
perpendicular. 

EXERCISES 

64.  The  center  of  a  circle  of  radius  4  inches  is  10  inches  from 
the  axis  about  which  it  rotates  to  form  a  torus.  Find  (1)  the  volume, 
(2)  the  surface  of  the  torus. 

Solution.    Applying  the  Theorem  of  Pappus,  we  have 
V  =  2  TrlO  •  TT¥=  320  -rr2  cubic  inches. 
S  =  2  TrlO  •  2  7r4  =  160  vr  square  inches. 

65.  How  many  cubic  feet  of  water  will  a  90°  elbow  of  an  8-inch 
water  main  contain  if  the  axis  about  which  the  generating  circle 
of  the  elbow  rotates  is  12  inches  from  the  center  of  the  circle  ? 

66.  How  much  will  an  iron  elbow  of  Exercise  65  weigh  if  the 
iron  is  one-half  inch  thick  ?    (A  cubic  foot  of  iron  weighs  about 
480  pounds.) 


BOOK  VII 


385 


67.  A  rectangle  2x4  inches  rotates  about  a  line  parallel  to 
the  nearer  one  of  the  shorter  sides  and  6  inches  from  it.    Find 
(1)  the  volume,  (2)  the  area  of  the  solid  generated. 

68.  Compare  the  results  of  the  foregoing  exercise  with  those 
obtained  if  the  same  rectangle  is  rotated  about  an  axis  6  inches 
from  the  longer  side. 

69.  The  cross  section  of  the  rim  of  an  iron  flywheel  3  feet  in 
diameter  is  a  rectangle  12  x  2  inches.    How  much  does  it  weigh  ? 

508.  Tangent  plane.  A  plane  which  contains  an  element  of 
a  cylindrical  surface  and  meets  the  surface  nowhere  else 
is  said  to  be  tangent  to  the  cylinder. 


509.  Circumscribed  prism.  A  prism  is  said  to  be  circum- 
scribed about  a  cylinder  when  its  bases  are  polygons  circum- 
scribed about  the  bases  of  the  cylinder  and  its  lateral  faces  are 
tangent  to  the  cylinder. 

EXERCISES 

70.  The  ratio  of  the  volume  of  a  cylinder  to  that  of  its  inscribed 
or  circumscribed  prisms  is  equal  to  the  ratio  of  their  correspond- 
ing bases. 

71.  A  cylinder  is  generated  by  the  revolution  of  a  rectangle 
2x6  inches  about  the  shorter  side.    Find  the  volume  and  the 
total  area. 

72.  A  cylinder  is  generated  by  the  revolution  of  a  rectangle 
2x6  inches  about  the  longer  side.    Find  the  volume  and  the 
total  area. 


386  SOLID  GEOMETRY 

73.  Two  cylinders  are  generated  by  the  revolution  of  a  rec- 
tangle a  x  b  inches,  first  about  the  side  a,  then  about  the  side  b. 
Find  the  ratio  (1)  of  the  volumes,  (2)  of  the  total  areas,  (3)  of  the 
lateral  areas,  of  the  cylinders  formed. 

74.  The  edge  of  a  cube  is  8  inches.    Find  the  volume  of  the 
circumscribed  cylinder. 

75.  The  lateral  area  of  a  cube  is  54  square  inches.    Find  the 
lateral  area  of  the  inscribed  cylinder. 

76.  A  tank  is  2  x  3  x  4  feet.    It  is  desired  to  make  a  cylindrical 
tin  tank,  without  a  lid,-  of  the  same  volume  and  having  an  altitude 
of  3  feet.    Find  the  number  of  square  feet  of  tin  in  the  new  tank. 

77.  A  cylindrical  gasoline  tank  48  inches  long  and  of  diameter 
16  inches  is  on  its  side  in  a  horizontal  position.    The  greatest 
depth  of  the  gasoline  is  1  foot.    How  many  gallons  are  required 
to  fill  the  tank  ?    (231  cubic  inches  =  1  gallon.) 

78.  Find  the  volume  of  a  cylinder  of  revolution  inscribed  in  a 
regular  hexagonal  prism  whose  altitude  is  8  inches  and  whose 
lateral  area  is  288. 

79.  Find  the  diameter  of  a  cylinder  of  revolution  if  the  volume 
is  equal  numerically  to  (1)  the  total  area,  (2)  the  lateral  area. 

80.  The  radius  of  a  cylinder  of  revolution  is  7  inches,  and  its 
altitude  is  15  inches.    Find  the  distance  from  the  axis  of  the  cylin- 
der to  the  plane  parallel  to  the  axis  which  makes  a  section  of  the 
cylinder  equal  in  area  to  the  base.    (Use  TT  =  22/7.) 

81.  What  is  the  ratio  of  the  diameter  to  the  altitude  of  a 
cylinder  of  revolution  if  the  area  of  the  greatest  section  made 
by  a  plane  through  an  element  is  equal  to  that  of  the  base  ? 

82.  A  cylinder  of  revolution  is  of  radius  8  inches.    Two  tan- 
gent planes  are  drawn  to  the  cylinder  through  a  point  8  inches 
from  the  surface  of  the  cylinder.    What  is  the  angle  between  the 
planes  ? 

83.  What  is  the  diameter  of  a  cylindrical  quart  measure  which 
is  4  inches  high  ? 


BOOK  VII 


387 


PYRAMIDS 

510.  Pyramid.  A  pyramid  is  the  solid  bounded  by  a 
polygon  and  triangles  having  the  sides  of  the  polygon  as 
their  bases  and  having  a  common 
vertex. 

The  polygon  is  called  the  base  of 
the  pyramid ;  the  triangles  are  called 
the  lateral  faces ;  and  the  common 
vertex  of  the  lateral  faces  is  called 
the  vertex. 

The  intersections  of  the  lateral  faces  are  called  the  lateral 
edges  of  the  pyramid. 

Pyramids  are  triangular,  quadrangular,  hexagonal,  etc.,  accord- 
ing as  their  bases  have  three,  four,  six,  etc.  sides. 


511.  Lateral  area.    The  sum  of  the  areas  of  the   lateral 
faces  is  the  lateral  area  of  the  pyramid. 

512.  Altitude.    The     altitude     of    a 
pyramid  is  the  perpendicular  distance 
from  the  vertex  to  the   plane   of  the 
base. 

QUERY  1.   Can  the  altitude  of  a  pyramid 
be  equal  to  one  of  the  lateral  edges? 

QUERY  2.  To  how  many  lateral  edges  can  the  altitude  of  a  pyramid 
be  equal? 


388 


SOLID  GEOMETRY 


QUERY  3.  For  what  kind  of  pyramid  may  any  one  of  the  faces  be 
taken  as  the  base  ? 

QUERY  4.  From  the  adjacent  figure 
name  the  various  parts  of  the  pyra- 
mid which  have  been  defined. 

QUERY  5.  How  many  pyramids  are 
there  with  a  given  base  and  a  given 
altitude?  What  is  the  locus  of  their 
vertices  ? 

QUERY  6.  As  the  vertex  of  a  pyra- 
mid becomes  very  remote  from  the 
base,  what  relation  to  each  other  do  the  lateral  edges  approach  ? 

513.  Pyramidal   surface.    The  lateral  faces  of  a  pyramid 
may  be  generated  by  the  motion  of  a  line  always  passing 
through  a  fixed  point,  the  vertex,  and  continually  meeting 
the    perimeter    of    a    fixed    polygon. 

If  we  consider  the  figure  generated 
by  the  entire  line,  we  have  not  merely 
the  lateral  surface  of  the  pyramid,  but 
a  surface  extending  indefinitely  in  both 
directions  from  the  vertex.  This  sur- 
face is  called  &  pyramidal  (py  ram 'I  dal) 
surface. 

The  terms  generator,  element,  and 
directrix  are  applied  to  the  pyramidal 
surface  in  the  same  sense  as  that  defined 
in  §  488. 

514.  Another  definition  of  a  pyramid. 
A  second  definition  of  a  pyramid  may 
be  given  which  is  identical  in  meaning 
with  that  of  §  510  but  which  is  more 
useful  in  dealing  with  certain  problems. 

A  pyramid  is  a  solid  bounded  by  a  pyramidal  surface  and 
a  plane  which  cuts  all  of  the  elements  on  the  same  side  of 
the  vertex. 


BOOK  VII 


389 


Theorem  11 


515.  If  a  pyramid  is  cut  by  a  plane  parallel  to  the  base, 

(a)  the  lateral  edges  and  the  altitude  are  divided  pro- 
portionally^ 

(b)  the  section  is  a  polygon  similar  to  the  base. 


Given  any  pyramid  0-CG,  cut  by  a  plane  M  parallel  to  the  base 
making  the  section  P-L.  Let  the  altitude  OS  intersect  M  at  the 
point  R. 

OH_OK_OL_         _OR 

}  ~OA~~OB~~OC~         ~~OS' 
(&)  P—L  is  a  polygon  similar  to  G~C. 

OH      OK       OL  OR 

Proof,  (a)  _  =  _  =  _=...  =  _.  §411 

(ft)  P#  is  II  to  (7/1;  HK  is  II  to  AB;  KL  is  II  to  BC\  etc.  Why? 

Therefore    Z  PHK  =  Z  GA  B ;  Z  HKL  =  Z  A  B  C,  etc.        Why  ? 

Also  AOAB  is  similar  to  AOHK-,  A  OB  C  is  similar  to  AOKL; 

etc.  §  271 

He-e         S-3' §'-.§!•*.'      §269 

Therefore  ^  =  —  5  etc.  Why? 

Hence          polygon  P-L  is  similar  to  polygon  G-C.  §  267 


390 


SOLID  GEOMETRY 


Theorem  12 


516.  If  two  pyramids  having  equal  bases  and  altitudes 
are  cut  by  planes  parallel  to  their  bases  and.  equidistant 
from  their  vertices,  the  sections  formed  are  equal. 


Given  any  two  pyramids,  with  equal  bases  -Band  B',  and  with 
equal  altitudes  h.  Let  them  be  cut  by  planes  parallel  to  the  bases 
and  distant  a  from  the  vertices.  Let  the  sections  be  called  S  and  S'. 


To  prove  that 

8  = 

8'. 

Proof. 

EF 

VF 

But 
Therefore 

VF 
VK~ 
EF 

a  OU 
h~  OR 
TU 

TU       OU 


Wh    ? 


EF' 


TU 


GK       PR '    "  -QK2       -pj? 
But  S  is  similar  to  B,  and  AS"  is  similar  to  B'. 


Hence 


Consequently 


EF 
GK' 


S' 


Why  ? 

§515  (ft) 

§326 

Why? 


But                              B  =  B '.  Given 

Therefore                     S  =  S'.  Why  ? 

QUERY.    In  Theorem  12,  under  what  circumstances  would  the  sec- 
tions made  by  the  planes  be  congruent? 


BOOK  VII 


391 


517.  Corollary.  If  a  pyramid  is  cut  by  two  parallel  planes, 
the  areas  of  the  sections  are  proportional  to  the  squares  of  their 
distances  from  the  vertex. 

QUERY  1.  If  a  plane  parallel  to  the  base  of  a  pyramid  is  halfway  be- 
tween the  base  and  the  vertex,  what  is  the  ratio  of  the  section  to  the  base  ? 

QUERY  2.    How  large  a  shadow  will  a  book  6x8  inches,  held  4  feet 
from  a  light,  cast  on  a  wall  parallel  to 
the  book  and  8  feet  away  from  the  light  ? 

QUERY  3.  How  much  larger  would 
the  shadow  in  Query  2  be  if  the  wall  of 
the  room  were  (a)  12  feet,  (b)  16  feet 
from  the  light? 

QUERY  4.  Explain  how  the  law  that 
the  intensity  of  a  light  varies  inversely 
with  the  square  of  its  distance  is  consistent  with  §  517. 

QUERY  5.  If  a  pyramidal  surface  is  cut  by  two  parallel  planes  on 
opposite  sides  of  the  vertex  (above  and  below  in  figure  of  §  513),  are  the 
sections  similar  ? 

QUERY  6.  If  a  pyramidal  surface  is  cut  by  two  parallel  planes  on 
opposite  sides  of  the  vertex  and  at  equal  distances  from  it,  are  the  sec- 
tions equal  ?  Are  they  congruent  ?  If  the  sections  are  triangles,  are 
they  congruent? 

EXERCISES 

84.  A  square  pyramid  has  all  of  its  lateral  faces  equilateral 
triangles  whose  sides  are  each  12  feet.    Find  the  altitude  of  the 
pyramid. 

85.  A  pyramid  of  altitude  6  feet  is  cut  by  a  plane  which  makes 
a  section  half  the  area  of  the  base.   How  far  from  the  vertex  of 
the  pyramid  is  the  plane  ? 

86.  If  a  pyramid  is  cut  by  two  parallel  planes,  the  correspond- 
ing sides  of  the  sections  are  in  the  same  ratio  as  the  distances  of 
the  planes  from  the  vertex. 

87.  If  two  pyramids  having  bases  of  equal  perimeter  and  equal 
altitudes  are  cut  by  planes  equidistant  from  their  vertices,  the 
sections  formed  have  equal  perimeters. 


392 


SOLID  GEOMETRY 
Theorem  13 


518.    Two  pyramids  having  equal  bases  and  equal  alti- 
tudes are  equal  in  volume. 


Given  any  two  pyramids  P  and  P',  with  equal  altitudes  h 
and  equal  bases  B  and  B'  in  the  same  plane. 

To  prove  that  P  =  P1. 

Proof.    At  any  distance  d  from  the  bases  pass  planes 
bases  of  the  pyramids,  making  sections  S  and  S'. 


to  the 


Then 

But 

Therefore 


d  is  any  distance  less  than  h. 
P  =  P'. 


§516 


§481 


QUKRY  1.    Is  Theorem  13  valid  for  two  pyramids  whose  bases  have 
different  numbers  of  sides  ? 

QUERY  2.    Is  the  converse  of  Theorem  13  true? 

EXERCISES 

88.  If   a  pyramidal    surface  is   cut   by  two   parallel   planes, 
each  cutting  all  the'  edges  on  opposite  sides  of  the  vertex  and 
at  equal  distances  from  it,  the  pyramids  formed  are  equal  in 
volume. 

89.  What  relation  do  the  two  plane  sections  of  Exercise  88 
bear  to  each  other  ?    Prove  your  statement. 


BOOK  VII 
Theorem  14 


393 


519.   The  volume  of  a  triangular  pyramid  equals  one 
third  of  the  product  of  its  base  and  altitude. 

R  0 


Given  any  triangular  pyramid  0-ABC,  with  altitude  h. 
To  prove  that  the  volume  0~ABC  =  1  h  X  ABC. 
Proof.      At  B  and  C  construct  BR  and  CS  II  to  OA. 

Pass  planes  determined  by  pairs  of  parallels  A O,  CS ;  A  0,  BR', 
BR,  CS. 

Pass  plane  ORS  through  0  II  to  ABC.  §  408 

The  complete  figure  formed  is  a  triangular  prism  having  ABC 
as  its  base  and  h  as  its  altitude.  Why  ? 

Pass  plane  ORC,  dividing  the  pyramid  O-RBCS  into  the  two 
triangular  pyramids  0-RCS  and  0-RBC. 
AABC  =  AORS. 

Therefore  0-ABC  =  C-RSO. 

Also  ARCB=ARCS. 

Therefore  O-RBC  =  0-RCS. 

But  0-RSC  —  C-RSO. 

Therefore  0-ABC  =  0-RCS  =  0-RBC. 

Hence  O-ABC  =  ^  prism  ABCORS. 

But  prism  ABCORS  =  hx  ABC. 

Therefore        volume  O-ABC  =  i  A  X  ^£C. 


Why? 

§518 

Why? 

Why? 

Identical 

Why? 

Why? 


394 


SOLID  GEOMETRY 


NOTE.  The  relation  between  the  three  pyramids  into  which  a  prism 
may  be  divided,  which  is  demonstrated  in  the  foregoing  theorem,  is 
found  in  Euclid's  Geometry  and  was  known  by  a  geometer  of  even  earlier 
date.  It  is  one  of  the  comparatively  few  theorems  in  our  modern  treat- 
ment of  solid  geometry  which  has  been  contained  in  nearly  all  the  text- 
books from  the  earliest  times  to  the  present.  The  Greeks  were  more 
interested  in  the  logic  of  geometry  than  in  its  application,  and  conse- 
quently paid  more  attention  to  plane  geometry  than  they  did  to  solid. 

Theorem  15 

520.  The  volume  of  any  pyramid  equals  one  third  of 
the  jiroduct  of  its  base  and  altitude. 


Given  any  pyramid  P,  with  base  B  and  altitude  h. 
To  prove  that        volume  P=  \  B  •  h. 

Proof.  From  any  vertex  V,  draw  all  the  diagonals  of  the  base 
B,  dividing  it  into  triangles  Blt  B2,  B^  etc. 

Pass  planes  determined  by  these  lines  of  division  and  the  vertex, 

dividing  the  pyramid  into  several  triangular  pyramids  P1?  Pa,  Pg,  etc. 

h  is  the  common  altitude  of  P1?  Pa,  P8,  etc.  §  512 

Now  P  =  i  B  .  h.  §  519 


Adding, 


•)> 


or 


P=$B.?i. 


BOOK  VII  395 

EXERCISES 

90.  Two  pyramids  have  bases  of  equal  areas,  but  one  is  14  feet 
high  and  the  other  is  10  feet  high.    Compare  their  volumes. 

91.  Two  pyramids  have  equal  altitudes.    The  base  of  one  is  a 
square  6  feet  on  a  side  ;  that  of  the  other  is  an  equilateral  triangle 
9  feet  on  a  side.    Compare  their  volumes. 

92.  Two  pyramids  have  the  same  altitude.    Their  bases  are 
respectively  a  square  and  a  regular  hexagon  which  can  be  in- 
scribed in  the  same  circle.    Compare  the  volumes. 

521.  Frustum  of  pyramid.  If  a  plane  is  passed  parallel  to 
the  base  of  a  pyramid  and  cutting  the  edges,  the  solid  in- 
cluded between  the  base  and  the  cutting  plane  is  called  a 
frustum  of  a  pyramid. 

The  section  formed  by  the  cutting  plane  is  called  the  upper 
base,  the  base  of  the  original  pyramid  is  called  the  lower  base,  and 
the  distance  between  the  bases  is  called  the  altitude,  of  the  frustum. 


In  case  the  cutting  plane  is  not  parallel  to  the  base  of  the 
pyramid,  but  cuts  all  of  the  lateral  edges,  the  solid  obtained 
is  called  a  truncated  pyramid. 

QUERY  1.  Why  are  the  bases  of  a  frustum  of  a  pyramid  similar 
polygons  ? 

QUERY  2.  How  many  lateral  faces  of  a  truncated  pyramid  may  be 
trapezoids  ? 

QUERY  3.  If  a  pyramid  with  a  base  a  few  inches  on  a  side  has  its 
vertex  10  miles  above  the  base,  what  would  a  frustum  of  the  pyramid 
near  the  base  resemble? 


396 


SOLID  GEOMETRY 


Theorem  16 


522.  The  volume  of  a  frustum  of  a  pyramid  equals  one 
third  of  the  product  of  the  altitude  and  sum  of  the  upper 
base,  the  lower  base,  and  the  mean  proportional  between 
the  two  bases. 


Given  F,  any  frustum  of  a  pyramid,  with  the  upper  base  U, 
the  lower  base  L,  and  the  altitude  h. 

To  prove  that  volume  F=  1  h  ( L  +  U  +  V  £7"  x  L  ). 

Proof.  Construct  the  pyramid  P,  of  which  F  is  a  frustum,  and 
let  its  altitude  be  h  +  x.  Denote  by  S  the  small  pyramid  which 
has  U  for  a  base  and  x  for  an  altitude. 

It  is  required  to  express  F  in  terms  of  L,  U,  and  h. 

Now  F  =  P  —  6'  =  ^  L (Ji  -\-  x)  —  ^  U  '  x  =  ^  (JiL  -+-  Lx  —  Ux) 

=  J  [h L  +  x(L-  C7)].  (1)          §  520 

We  must  now  eliminate  x,  that  is,  express  x  in  terms  of  L,  U, 
and  h,  and  substitute  the  value  obtained  in  (1). 

x2  U 


WTW  = 

Extracting  the  square  root, 


§517 


Vu 

vz 


Solving  this  equation  for  x, 


x  = 


BOOK  VII  397 

Substituting  in  (1),  we  obtain 

F= 


=  i-  \1iL  +  h 
(since  L-  U  =  (  VI  -  Vu)(  VZ+  VZ/)) 


NOTE.  The  foregoing  rule  for  the  volume  of  the  frustum  of  a  pyra- 
mid was  first  given  by  Heron  of  Alexandria  in  substantially  the  same 
form  as  in  this  text.  A  Hindu  mathematician  named  Brahmagupta 
(about  650)  gave  a  rule  for  the  volume  of  the  frustum  of  a  pyramid 
with  square  bases  of  sides  S1  and  S2  as  follows  : 


QUERY  1.  A  pyramid  may  be  considered  as  a  frustum  of  a  pyramid 
with  its  upper  base  equal  to  zero.  Verify  the  formula  for  the  volume 
of  a  frustum  of  a  pyramid  in  this  case. 

QUERY  2.  Verify  the  formula  for  the  frustum 
of  a  pyramid  for  the  case  where  the  upper  base 
equals  the  lower  base.  What  is  a  more  familiar 
name  for  this  solid  ? 

523.  Tetrahedron.  A  polyhedron  hav- 
ing four  faces  is  called  a  tetrahedron.  The 
terms  tetrahedron  and  triangular  pyramid  are  interchangeable. 

Two  skew  edges  of  a  tetrahedron  are  called  opposite  edges. 

EXERCISES 

93.  If  the  base  of  a  regular  pyramid  10  feet  high  contains  16 
square  feet,  what  is  its  volume  ? 

94.  What  is  the  volume  of  a  square  pyramid  12  feet  high  if 
the  base  is  6  feet  on  a  side  ? 

95.  The  base  of  a  pyramid  is  an  equilateral  triangle  6  inches 
on  a  side,  and  its  altitude  is  8  inches.    Find  the  volume. 


398  SOLID  GEOMETRY 

96.  What  is  the  volume  of  the  frustum  of  a  pyramid  whose 
bases  are  equilateral  triangles,  3  feet  and  6  feet  on  a  side  respec- 
tively, and  whose  altitude  is  5  feet  ? 

97.  The  altitude  of  a  pyramid  with  a  square  base  is  8  inches. 
The  volume  is  128  cubic  inches.    Find  a  side  of  the  base. 

98.  The  base  of  a  pyramid  is  an  isosceles  triangle  whose  sides 
are  10, 10,  and  6  inches.    The  altitude  of  the  pyramid  is  15  inches. 
Find  the  volume. 

99.  The  sides  of  the  base  of  a  tetrahedron  are  10,  17,  and  21 
inches,  and  its  altitude  is  6  inches.    Find  its  volume. 

100.  The  base  of  a  pyramid  is  twice  that  of  a  section  parallel  to 
it  and  3  feet  above  it.    Find  the  altitude  of  the  pyramid. 

101.  A  pyramid  is  18  inches  in  height.    A  section  1  foot  from 
the  base  and  parallel  to  it  has  an  area  of  56  square  inches.   Find 
the  volume  of  the  pyramid. 

102.  The  base  of  a  pyramid  contains  64  square  inches,  and  its 
altitude  is  8  inches.    How  far  from  the  vertex  must  a  plane  par- 
allel to  the  base  be  passed  in  order  to  get  a  section  one  quarter 
of  the  area  of  the  base? 

103.  A  frustum  of  a  pyramid  whose   square   lower   base   is 
40  feet  on  a  side  consists  of  earth  which  rests  at  an  angle  of  45°. 
How  high  is  the  frustum  if  the  top  contains  100  square  feet? 
How  many  cubic  feet  are  there  in  the  frustum  ? 

104.  A  miller  wishes  to  make  a  hopper  in  the  form  of  an  in- 
verted frustum  of  a  square  pyramid,  with  the  bases  6  and  3  feet 
en  a  side,  respectively.    How  deep  shall  he  make  it  in  order  that 
it  shall  hold  15  bushels  of  grain  ?    (A  bushel  =  1.24  cubic  feet.) 

105.  Cleopatra's  Needle,  the  Egyptian  obelisk  in  New  York 
City,  consists  of  a  frustum  of  a  pyramid  whose  lower  base  is 
a  square   1\  feet  on  a  side,  whose  upper  base  is  4  feet  on  a 
side,  and  whose  altitude  is  61  feet,  surmounted  by  a  pyramid 
whose  base  is  the  upper  base  of  the  frustum  and  whose  altitude 
is  1\  feet.    Find  the  weight  of  the  Needle  in  tons.    (One  cubic 
foot  of  the  stone  weighs  about  170  pounds.) 


BOOK  VII  399 

524.  Regular  pyramid.    A  regular  pyramid  is  one  whose 
base  is    a   regular  polygon    and  whose   vertex    lies    in    the 
perpendicular  erected  at  the  center  of  the 

base. 

(The  center  of  a  regular  polygon  is  the 
center  of  its  circumscribing  circle.) 

EXERCISE  106.  Prove  that  the  perpendicular 
from  the  vertex  to  the  base  of  a  regular  pyramid 
passes  through  the  center  of  the  circle  which 
circumscribes  the  base. 

525.  Regular  tetrahedron.    A  regular  triangular  pyramid 
has    an  equilateral    triangle  as   base   (§  524).    Its   altitude 
may  have  any  length.    There  may  be,  therefore,  any  number  of 
regular  triangular  pyramids  on  the  same  base. 

The  regular  triangular  pyramid  which  has  all 
its  lateral  faces  equilateral  triangles  is  called  a 
regular  tetrahedron. 

QUERY.  Are  the  lateral  faces  of  a  regular  tetra- 
hedron congruent  to  the  base  ? 

Prove    each    of    the    following  propositions   concerning  a 
regular  pyramid: 

526.  The  lateral  edges  of  a  regular  pyramid  are  equal. 

527.  The  lateral  faces  of  a  regular  pyramid  are  congruent 
isosceles  triangles. 

528.  The  altitudes  of  the  lateral  faces  of  a  regular  pyramid 
are  equal. 

529.  If  a  regular  pyramid  is  cut  by  a  plane  parallel  .to  its 
base,  the  pyramid  cut  off  has  equal  lateral  edges. 

530.  Slant  height.    The  slant  height  of  a  regular  pyramid 
is  the  altitude  of  any  of  its  lateral  faces. 


400  SOLID  GEOMETRY 

Theorem  17 

531.  The  lateral  area  of  a  regular  pyramid  equals 
the  product  of  one  half  the  perimeter  of  the  base  and  the 
slant  height. 


Given  any  regular  pyramid  P,  with  the  base  ABCDFG  and 
the  slant  height  /. 

To  prove    lateral  area  of  P  =  1 1 .  perimeter  AB  CDFG. 
Proof.  Area  of  face  OAB=%l*AB.  §  320 

Area  of  face  OBC  =  %l .  EC.  §  528 


Adding  and  factoring, 

Area (OAB  +  OBC  -\ )=  %l(AB  +  BC  -\ ), 

or  lateral  area  P  =  %  I  -  perimeter  ABCDFG. 

QUERY  1.  Does  a  pyramid  which  is  not  regular  have  any  slant  height  ? 
How  could  its  lateral  area  be  found? 

QUERY  2.  If  two  regular  pyramids,  one  having  a  square  base  and  the 
other  having  a  hexagonal  base,  have  equal  altitudes,  and  if  their  bases 
are  inscribed  in  the  same  circle,  which  one  has  the  greater  slant  height  ? 

QUERY  3.  If  two  regular  pyramids  have  equal  altitudes,  and  bases 
of  equal  area  but  of  different  numbers  of  sides,  are  their  lateral  areas 
equal?.  Discuss  completely. 

QUERY  4.  If  two  pyramids  have  equal  lateral  areas,  are  their  volumes 
necessarily  equal? 

QUERY  5.  Can  a  pyramid  with  very  small  lateral  area  have  a  great 
volume?  Can  one  with  a  small  volume  have  a  great  lateral  area? 


BOOK  VII        :    :,,,,,,  401 

Theorem  18 

532.  The  laleral  faces  of  a  frustum  of  a  'regular 'pi/rk^ 
mid  are  congruent  trapezoids. 


Given  a  frustum  F  of  a  regular  pyramid,  and  let  ABDC 
and  SKHR  be  any  two  lateral  faces. 

To  prove  that  ABDC  and  SKHR  are  congruent  trapezoids. 
Proof.  RH  is  II  to  SK  and  CD  is  II  to  AR.  Why  ? 

Hence  ^££(7  and  SKHR  are  trapezoids.  §  107 

Let  0  be  the  vertex  of  the  pyramid. 

To  prove  ABDC  and  KACH  congruent,  fold  over  OAK  on  OA 
as  axis,  so  that  it  coincides  with  OAB.  §  20 

Then  AK  coincides  with  AB,  §§  79,  527 

and  therefore  CH  coincides  with  CD.  §  45 

Therefore   ABDC  and  KACH  coincide  throughout. 

Similarly,  any  lateral  face  can  be  proved  congruent  to  an  adja- 
cent one,  and  hence  to  any  lateral  face  of  the  frustum.       §  32  (2) 

533.  Slant  height  of  frustum.    The  slant  height  of  a  frustum 
of  a  regular  pyramid  is  the  altitude  of  any  of  its  faces. 

QUERY  1.  Would  a  truncated  pyramid  have  any  slant  height? 
QUERY  2.  Would  the  frustum  of  a  pyramid  which  is  not  regular 
have  any  slant  height? 


402 


SOLID  GEOMETRY 


^....    .  /,  •      Theorem  19 

••'•"•  ti&::T]ie  lateral  area  of  the  frustum  of  a  regular  pyra- 
mid is  equal  to  one  half  the  product  of  the  sum  of  the 
perimeters  of  the  bases  and  the  slant  height. 
Proof  is  left  to  the  student. 

EXERCISES 

107.  The  lateral  area  of  a  pyramid  is  greater  than  the  area  of 
its  base. 

108.  If  the  regular  tetrahedron  of  the  adja- 
cent figure  has  an  edge  6  inches  long,  compute 
the  lines  AL,  KL,  AR,  and  KR,  where  L  is 
the  mid-point  of  CB  and  KR  is  the  altitude 
of  the  tetrahedron. 

HINT.    See  Exercise  106,  page  399. 

109.  If  the  frustum  of  a  regular  pyramid 
in  the  adjacent  figure  has   slant  height  10 
and  sides  of  the  bases  24  and  12  respectively, 
compute  RA,  KA,  AL,  where  KR  is  the  slant 
height  and  KA  the  altitude  of  the  frustum. 

In  Exercises  110-116,  all  of  which  relate 
to  pyramids,  li  denotes  the  altitude,  B  the  area  of  the  base,  b  one 
side  of  the  base,  n  the  number  of  sides  of  the  base,  I  the  slant 
height,  S  the  lateral  area,  and  V  the  volume. 

110.  Given  F=  46,  h  =113/4.    Find  B. 

111.  Given  B  =  12,  A  =  15.    Find  V. 

In  Exercises  112-116  the  pyramid  is  regular. 

112.  Given  b  =  6,  n  =  4,  h  =  4.    Find  S. 

113.  Given  V=  25,  h  =  3,  n  =  4.    Find  S. 

114.  Given  S  =  100,  b  =  4,  n  =  5.    Find  I. 

115.  Given  I  =  4,  n  =  3,  b  =  6,    Find  A, 


BOOK  VII  403 

116.  Given  n=  3,  £  =  16, 1=5.   Find  T, 

117.  The  area  of  the  base  of  a  regular  quadrangular'  pyr.'inn-l 
is  36  square  inches.    A  lateral  edge  is  10  inches.    Find  (1)  the 
lateral  area,  (2)  the  volume. 

118.  One  side  of  the  base  of  a  regular  tetrahedron  is  8  feet. 
Find  the  lateral  area. 

119.  The  frustum  of  a  regular  quadrangular  pyramid  has  bases 
with  edges  12  and  18  respectively.    Its  altitude  is  4.    Find  the 
lateral  area. 

120.  The  frustum  of  a  regular  triangular  pyramid  has  bases 
with  sides  6  and  12  inches  respectively,  and  an  altitude  of  4  inches. 
Find  the  lateral  area. 

121.  A  regular  quadrangular  pyramid  has  an  altitude  of  12  feet, 
and  a  base  24  feet  on  a  side.    Find  the  angle  between  a  face  and 
the  base. 

122.  A  regular  quadrangular  pyramid  has  a  base  6  inches  on  a 
side,  and  a  lateral  face  makes  an  angle  of  60°  with  the  base.    Find 
(1)  the  lateral  area,  (2)  the  volume. 

123.  It  is  desired  to  make  a  regular  triangular  pyramid  equal 
in  volume  and  altitude  to  a  given  regular  hexagonal  pyramid 
whose  base  is  6  feet  on  a  side.    How  long  is  a  side  of  the  base  of 
the  new  pyramid  ? 

124.  The  bases  of  a  frustum  of  a  regular  pyramid  are  equi- 
lateral triangles  whose  sides  are  4  and  6  inches  respectively. 
The  slant  height  of  the  frustum  is  12  inches.    Find  (1)  the  lateral 
area  and  (2)  the  total  area  of  the  frustum. 

125.  Prove  that  the  altitude  of  a  regular  tetrahedron  is  1  V6 
times  the  edge. 

HINT.    The  altitude  meets  the  base  at  the  intersection  of  its  medians. 

126.  Prove  that  for  a  regular  tetrahedron  whose  edge  is  a  the 

a*V2 
volume  is  equal  to  • 


4.Q4  .          ••::•::  SOLID  GEOMETRY 


127..  t.Lf  t the.  base  of/a  pyramid  is  a  parallelogram,  and  the  section 
byii-I^arie  not?' parallel  to  the  base  has  one  side  parallel  to 
one  side  of  the  base,  prove  that  the  section  is  a  trapezoid. 

128.  If  each  pair  of  opposite  edges  of  a  given  tetrahedron 
are   equal,   and  the    surface   of   the    tetrahedron    is    cut   along 
three    concurrent    edges    and   opened    put 

flat,  prove    that   the    figure   formed    is   a 
triangle. 

129.  How  many  right  angles  in  the  sum 
of  the  angles  of  all  the  faces  of  a  pyramid 
whose  base  has  n  sides  ? 

130.  In  a  tetrahedron  0-ABC  the  points 
P,  R,  S,  T,  which  bisect  the  edges  OA,  OB, 
BC,  AC,  are  coplanar. 

131.  In  a  tetrahedron,  if  a  plane  is  passed  parallel  to  two  oppo- 
site edges,  the  section  is  a  parallelogram. 

132.  In  the  tetrahedron   0-ABC  let  D  and  F  be  the  points 
where  the  medians    of   the    faces    OBC  and  ABC   respectively 
meet.   Prove  (1)  that  the  lines  AD  and  OF 

meet,  (2)  that  the  triangle  R  OA  is  similar 
to  RFD. 

133.  Prove  that  the  three  lines  drawn 
from  the  vertices  of  a  tetrahedron  to  the 
intersections  of  the  medians  of  the  oppo- 
site faces  meet  in  a  point  which  divides 
each  of  the  lines  in  the  ratio  3 : 1. 

These  lines  are  called  the  medians  of 
the  tetrahedron. 

NOTE.  The  point  located  in  Exercise  133  is  called  the  center  of 
gravity  of  the  tetrahedron.  It  is  the  point  about  which  the  solid  would 
exactly  balance  if  it  could  be  supported  at  that  point.  It  is  interesting 
to  note  that  the  center  of  gravity  of  a  triangle  divides  the  medians  in 
the  ratio  2  :1,  while  that  of  the  tetrahedron  divides  the  medians  in  the 
ratio  3:1. 


BOOK  VII 


405 


CONES 

535.  Conical  surface.    A  conical  surface  is  generated  by  a 
moving    line    which    always    passes    through   a 

given  fixed  point  and  continually  intersects 
a  given  fixed  curve. 

The  fixed  point  is  called  the  vertex,  the  fixed 
curve  the  directrix,  and  the  moving  line  the 
generator,  of  the  conical  surface. 

The  surface  consists  of  two  parts,  or  nappes, 
connected  only  at  the  vertex. 

536.  Cone.    The  solid  bounded  by  a  conical  surface  and  a 
plane  whose  section  with  the   surface  is  a  closed  curve  is 
called  a  cone. 

In  this  text  only  convex  cones,  that  is, 
cones  whose  directrices  are  convex  curves, 
are  treated. 

The  terms  base,  altitude,  and  frustum  are 
defined  similarly  to  the  corresponding  terms 
applied  to  the  pyramid. 

537.  Element.    The  generator  of  a  conical  surface  in  any 
of  its  positions  is  called  an  element  of  the  conical  surface. 

There  is  one  and  only  one  element  containing  the  vertex 
and  a  given  point  of  the  directrix. 

538.  Element  of  a  cone.    The  segment  of  the  element  of  a 
conical  surface  included  between  the  vertex  of  the  cone  and 
the  base  is  called  an  element  of  the  cone. 

QUERY  1.  What  kind  of  surface  would  be  generated  if  the  vertex 
and  the  directrix  of  a  conical  surface  were  in  the  same  plane  ? 

QUERY  2.  What  is  the  locus  of  lines  through  a  given  point  which 
make  a  given  angle  with  a  given  plane  ? 

QUERY  3.  What  is  the  locus  of  lines  making  a  given  angle  with  a 
given  plane  at  a  given  point?  What  if  the  given  angle  is  90°? 

QUERY  4.    Define  base,  altitude,  and  frustum  as  applied  to  the  cone. 


406 


SOLID  GEOMETRY 
Theorem  20 


539.  If  one  of  two  parallel  planes  intersects  a  conical 
surface  in  a  circle,  the  other  does  also,  and  the  vertex  of 
the  conical  surface  is  in  a  straight  line  with  the  centers  of 
the  two  circles. 


Given  a  conical  surface  with  vertex  0  cut  by  the  two  paral- 
lel planes  M  and  N.  Let  the  intersection  with  N  be  a  circle  with 
center  X.  Let  OX  intersect  N  at  the  point  P. 

To  prove  that  the  section  with  M  is  a  circle  whose  center  is  P. 

Proof.  Let  OF  and  OH  be  the  elements  through  any  two  points 
R  and  S  respectively  on  the  section.  Pass  planes  determined  by 
XO  and  FO,  and  by  XO  and  //(>,  intersecting  N  in  XF  and  XH, 
and  M  in  PR  and  PS,  respectively. 

XF  is  II  to  PR  ;   XH  is  II  to  PS. 

PS^_OP_.  PR  _  OP 

XH~  OX'    XF~  OX' 
PS       PR 


Why? 
271,  269 


Therefore 

But 

Therefore 


XH  =  XF. 
PS  =  PR. 


But  R  and  S  were  any  points  on  the  section. 
Therefore  the  section  is  circular,  with  P  as  its  center. 


Why? 

Why? 
Why? 


Why? 


BOOK  VII 


40T 


540.  Corollary.  If  either  base  of  a  frustum  of  a  cone  is  cir- 
cular, both  bases  are  circular. 

NOTE.  It  appears  from  Theorem  20  that  if  a  plane  intersects  a  conical 
surface  in  a  circle,  all  the  planes  which  are  parallel  to  it  also  cut  the 
surface  in  circles.  All  other  planes  cut  the  surface  in  curves  which 
are  called  conic  sections.  The  names  of  these  conic  sections  are  the 
ellipse,  the  hyperbola,  and  the  parabola. 

Of  these  the  ellipse  is  closed,  while  the  others  are  open  curves. 
If  the  cutting  plane  is  not  quite  parallel  to  the  circular  base  of  the 


cone,  the  section  is  a  closed  curve  a  little  longer  in  one  direction  than 
in  another.  If  the  plane  is  passed  near  the  vertex,  and  barely  cutting 
all  the  elements,  a  very  narrow  closed  curve  is  obtained.  Any  one  of 
the  closed  sections  of  a  conical  surface  whose  directrix  is  a  circle  is 
called  an  ellipse,  except,  of  course,  those  that  are  circular. 

If  the  cutting  plane  is  parallel  to  one  of  the  elements,  it  cuts  all  the 
elements  except  this  one,  and  hence  is  an  open  curve,  called  a  parabola. 

If  the  plane  cuts  both  nappes  of  the  conical  surface,  making  a  curve 
with  two  separate  branches,  it  is  called  a  hyperbola. 

The  Greeks  studied  the  properties  of  all  the  conic  sections  by 
methods  similar  to  those  of  this  text,  and  derived  many  important 
properties.  These  results  were  regarded  as  mathematical  curiosities 
without  any  application  to  nature  until  Kepler  (1571-1630)  discovered 
that  the  planets  move  around  the  sun  in  ellipses,  and  that  the  properties 


408  SOLID  GEOMETRY 

of   the  curves   discovered   so  long  before   were  useful  in  explaining 
their  motions. 

Since  the  time  of  Kepler  thousands  of  applications  of  the  properties 
of  the  conic  sections  have  been  discovered  in  all  branches  of  science,  and 
a  thorough  knowledge  of  them  is  an  important  part  of  the  equipment  of 
the  modern  scientist.  This  is  acquired  nowadays  most  effectively  through 
the  study  of  analytic  geometry,  a  branch  of  mathemetics  whose  early 
development  was  due  to  the  French  philosopher  Descartes  (1596-1650). 
The  fundamental  ideas  of  analytics  are  now  usually  taught  in  connec- 
tion with  algebra  and  are  included  in  the  work  on  graphs.  But  the 
complete  elaboration  of  the  subject  includes  extensive  studies  of  all 
kinds  of  curves  and  surfaces,  of  which  the  simplest  and  the  most  useful 
are  the  conic  sections. 

541.  Circular  Cone.    A  circular  cone  is  one  which  has  a  cir- 
cular section  such  that  the  line  joining  the  vertex  of  the  cone 
to  the  center  of  the  circle  is  perpendicular 

to  the  plane  of  the  circle. 

A  circular  cone  does  not  necessarily  have 
a  circular  base. 

In  the  adjacent  figure  the  plane  M 
makes  a  circular  section  of  the  cone  of 
which  T  is  the  center,  and  OT  is  perpendicular  to  M. 

542.  Right  circular  cone.    A  right  circular  cone  is  a  circular 
cone  with  a  circular  base. 

543.  Axis.    The  line  joining  the  vertex  of  a  right  circular 
cone  with  the  center  of  its  base  is  called  the  axis 

of  the  cone. 

544.  Corollary.    The  elements  of  a  right  circular 
cone  are  equal. 

HINT.   Apply  §  425. 

545.  Slant  height.    The  length  of  an  element  of  a  right 
circular  cone  is  called  its  slant  height. 


BOOK  VII 
Theorem  21 


409 


546.    The  section  of  a  cone  made  by  a  plane  which 
passes  through  the  vertex  and  cuts  the  base  is  a  triangle. 


Given  any  cone  O-KR,  and  a  plane  which  contains  the  vertex 
and  cuts  the  base  in  AB. 

To  prove  that       the  section  OAB  is  a  triangle. 

Proof.                         AB  is  a  straight  line.  Why? 
The  straight  line  joining  0  and  A  lies  in  the  given  plane   §  381 

and  also  in  the  conical  surface.  §  535 

Therefore  the  intersection  OA  is  this  straight  line.  §  390 
Similarly,                   OB  is  a  straight  line. 

Therefore           the  section  A  OB  is  a  triangle.  §  16 

547.  Inscribed  pyramid.  A  pyramid  whose  base  is  inscribed 
in  the  base  of  a  given  cone,  and  whose  vertex  is  the  vertex 
of  the  cone,  is  said  to  be  inscribed  in  the  cone. 

QUERY  1.  If  the  vertex  of  a  cone  is  remote  from  the  base, 
a  frustum  of  the  cone  near  the  base  resembles  what  solid  ? 

QUERY  2.    To  what  theorem  regarding  the  cylinder 
does  Theorem  21  correspond? 

QUERY  3.   Why  will  the  lateral  edges  of  an  inscribed 
pyramid  be  elements  of  the  cone  ? 


410  SOLID  GEOMETRY 

QUERY  4.  Will  the  slant  height  of  an  inscribed  pyramid  ever  be 
equal  to  an  element  of  the  cone  ? 

QUERY  5.  Which  will  be  greater,  the  slant  height  of  a  regular  tri- 
angular pyramid  or  that  of  a  quadrangular  pyramid,  if  both  are  inscribed 
in  the  same  cone  ? 

QUERY  6.  If  a  pyramid  is  inscribed  in  a  right  circular  cone,  what 
kind  of  triangles  form  its  lateral  faces? 

QUERY  7.  Will  the  altitudes  of  all  the  triangles  mentioned  in  Query  6 
necessarily  be  equal? 

EXERCISES 

134.  Construct  a  regular  hexagonal  pyramid  inscribed  in  a  right 
circular  cone,  giving  a  reason  for  each  step. 

135.  Given  a  right  circular  cone  with  altitude  8  and  radius  6, 
find  the  following  for  the  inscribed  regular  quadrangular  pyramid  : 
(1)  area  of  base,  (2)  slant  height,  (3)  volume,  (4)  lateral  area. 

136.  Find  the  parts  required  in  Exercise  135  for  a  regular 
hexagonal  pyramid  inscribed  in  the  same  cone. 

548.  Volume  of  cone.  If  the  base  of  a  pyramid  which  is 
inscribed  in  a  cone  has  a  very  large  number  of  sides,  so  that 
each  side  is  very  short,  then  the  area  of  the  base  of  the  pyra- 
mid is  approximately  equal  to  the  area  of  the  base  of  the  cone. 
By  taking  the  sides  sufficiently  short,  thereby  increasing  their 
number,  as  close  an  approximation  as  desired  may  be  obtained. 

The  volume  of  the  inscribed  pyramid  whose  base  is  almost 
the  same  in  area  as  that  of  the  circumscribed  cone  will  differ 
very  little  from  that  of  the  cone.  But  the  volume  of  the 
pyramid  is  one  third  the  product  of  its  base  and  altitude, 
however  many  lateral  faces  it  may  have.  Since  the  volume, 
and  the  area  of  the  base,  of  the  cone  can  be  made  to  differ 
as  little  as  we  may  desire  from  the  same  features  of  the  pyra- 
mid, and  since  their  altitudes  are  identical,  we  shall  make  the 
statements  of  theorems  22,  23,  and  24  without  proof.  They 
can  be  rigorously  demonstrated  by  the  theory  of  limits. 


BOOK  VII  411 

Theorem  22 

549.  The  volume  of  a  cone  is  equal  to  one  third  the 
product  of  the  area  of  its  base  and  its  altitude. 

550.  Corollary.    The  volume,  V,  of  a  cone  with  a  circular  base  is 

V=±7rr% 
where  r  is  the  radius  of  the  base  and  h  is  the  altitude  of  the  cone. 

Theorem  23 

551.  The  volume,  V,  of  a  frustum  of  a  cone  is 


where  h  is  the  altitude  and  L  and  U  the  areas  of  the  lower 
and  upper  bases  respectively  of  the  frustum. 

552.  Corollary.    The  volume,  V,  of  a  frustum  of  a  circular  cone  is 


where  h  is  the  altitude  and  r  and  rl  are  the  radii  of  the  lower 
and  upper  bases  respectively  of  the  frmtum. 

HINT.    Apply  §  551. 

Theorem  24 

553.  The  volumes  of  two  prisms,  cylinders,  pyramids,  or 
cones 

(1)  are  to  each  other  as  the  products  of  their  bases  and 
altitudes  ; 

(2)  having  equalbases  are  to  each  other  as  their  altitudes; 

(3)  having  equal  altitudes  are  to  each  other  as  their  bases. 

HINT.  Denote  the  altitude  by  h  and  the  base  by  B  for  each  solid, 
and  use  the  appropriate  formula  for  the  volume. 

QUERY  1.  What  is  the  ratio  of  the  volume  of  a  cylinder  to  that  of 
a  cone  with  the  same  base  and  altitude  ? 


412  SOLID  GEOMETRY 

QUERY  2.  If  two  cones  have  equal  bases  and  equal  slant  heights,  are 
their  volumes  necessarily  equal  ? 

QUERY  3.  What  is  the  locus  of  the  vertices  of  cones  which  have  the 
same  base  and  equal  altitudes? 

QUERY  4.   Does  every  cone  have  a  slant  height?   Explain. 

EXERCISES 

137.  The  altitude  of  a  right  circular  cone  is  16  feet,  and  its 
slant  height  is  29  feet.    Find  its  volume. 

138.  What  is  the  radius  of  the  base  of  a  circular  cone  whose 
volume  is  100  cubic  inches  and  whose  altitude  is  5  inches  ? 

139.  The  radius  of  a  right  circular  cone  is  6  inches.     Find 
the  area  of  the  section  formed  by  a  plane  containing  the  axis  of 
the  cone  if  the  element  is  12  inches  long. 

140.  A  right  circular  cone  with  altitude  10  inches  and  radius 
6  inches  is  cut  by  a  plane  through  the  vertex  so  as  to  make  a 
section  having  an  area  just  half  as  great  as  that  of  the  greatest 
triangular  section.    How  far  from  the  center  of  the  base  is  the 
intersection  of  this  plane  with  the  base  ? 

141.  The  frustum  of  a  right  circular  cone  has  an  altitude  of 
12  inches  and  radii  of  4  and  6  inches  respectively.    Find  its  volume. 

142.  The  slant  height  of  a  frustum  of  a  right  circular  cone  is 
20  feet.    The  radii  are  2  and  8  feet  respectively.    Find  the  volume. 

554.  Lateral  area  of  a  cone.  The  lateral  area  of  a  cone  is 
the  area  of  the  conical  surface  of  the  cone. 

We  cannot  conveniently  apply  the  unit  of  area  to  such  a 
curved  surface  as  a  conical  surface  so  as  to  estimate  the 
number  of  times  it  is  contained  in  the  surface.  Yet  the  fact 
that  some  number  exists  which  tells  how  many  times  the  unit 
area  is  contained  in  the  conical  surface  is  so  clear  that  no 
one  but  a  student  of  advanced  mathematics  would  ever  think 
of  requiring  a  proof  of  the  statement.  Such  discussions  would 
be  out  of  place  here. 


BOOK  VII  413 

Since  the  lateral  area  of  a  regular  pyramid  equals  one  half 
the  product  of  its  slant  height  and  the  perimeter  of  its  base, 
whether  there  are  few  or  many  lateral  faces,  and  since,  when 
the  lateral  faces  become  very  numerous  and  at  the  same  time 
very  narrow,  the  lateral  area,  the  perimeter  of  the  base,  and  the 
slant  height,  of  a  regular  pyramid  inscribed  in  a  right  circular 
cone  are  approximately  equal  to  the  same  features  of  the  cone, 
we  may  make  the  following  statements,  which  will  be  assumed 
without  proof. 

Theorem  25 

555.  The  lateral  area  of  a  right  circular  cone  is  equal 
to  one  half  the  product  of  the  perimeter  of  the  base  and 
the  slant  height. 

Expressed  in  symbols, 

2  -rrrl 


where  S  is  the  lateral  area,  r  is  the  radius,  and  I  is  the  slant 
height,  of  the  cone. 

NOTE.  The  student 
can  further  convince 
himself  of  the  truth 
of  this  formula  by  roll- 
ing about  its  vertex 
and  on  a  plane  surface 
a  right  circular  none 
which  rests  on  one  of 
its  elements.  If  the 
cone  is  rolled  contin- 
uously in  the  same 
direction  until  the  ele- 
ment comes  in  contact 

with  the  surface  for  the  second  time,  it  will  appear  that  the  part  of 
the   plane  with  which  the  cone  has  come  in  contact  is  a  sector  of  a 


414  SOLID  GEOMETRY 

circle  the  arc  of  which  equals  2  irr,  the  perimeter  of  the  base  of  the 

cone,  and  the  radius  of  which  equals  /,  the  slant  height  of  the  cone. 

Now  the  area  of  this  sector  equals  one  half  of  the  product  of  the  arc 

which  forms  its  base,  2  TTT,  and  the  radius,  /,  of  the  circle  (§  369).    Here 

again  we  have  2  TTT  •  I 

S  =       -       =  irrl. 

556.  Corollary.    The  lateral  area  of  a  right  circular  cone  is 
equal  to  the  product  of  the  slant  height  and  the  perimeter  of 
the  circle  halfway  between  the  base  and  the  vertex. 

557.  Lateral  area  of  frustum  of  cone.    Since  the  lateral 
area  of  the  frustum  of  a  regular  pyramid  equals  one  half  the 
product  of  the  sum  of  the  perimeters  of  the  bases  and  the 
slant  height,  whether  there  are  few  or  many  equal  isosceles 
trapezoids  as  lateral  faces,  and  since,  when  these  trapezoids 
become  very  numerous  and  at  the  same  time  very  narrow,  the 
lateral  area,  the  perimeter  of  the  base,  and  the  slant  height 
of  the  frustum  of  a  regular  pyramid  inscribed  in  the  frustum 
of    a   right   circular    cone    are    approximately  equal  to   the 
same  features  of  the  frustum  of  the*  cone,  we  may  assume 
the  following  : 

Theorem  26 

558.  The  lateral  area  of  a  frustum  of  a  right  circular 
cone  is  equal  to  one  half  the  product  of  the  sum  of  the 
perimeters  of  the  bases  times  the  slant  height. 

Expressed  in  symbols, 


where  S  is  the  lateral  area,  I  the  slant  height,  r  and  rl  the 
radii  of  the  bases  of  the  frustum. 

NOTE.  Rolling  a  frustum  of  a  cone  in  the  manner  described  in  the  pre- 
ceding note  shows  that  the  lateral  area  of  the  frustum  equals  an  incomplete 
sector  of  a  circle. 


BOOK  VII 


415 


Let  x  denote  the  portion  of  the  slant  height  between  the  upper  base 
of  the  frustum  and  the  vertex  of  the  completed  cone. 

Then  r  =  ~  « 


Area  rolled  out  =  TTT  (/  +  #)  —  irr^x  =  irrl  +  irrx  — 
From  (1),  rx  =  r^l  +  r^x. 

Substituting  in  (2)  we  obtain, 

area  of  frustum  =  irrl  +  irr^l  =  Trl  (r  +  r^. 
Hence  S  =  7rl(r  +  rt). 


(2) 


559.  Corollary.  The  lateral  area  of  a  frustum  of  a  right 
circular  cone  is  equal  to  the  product  of  the  slant  height  by 
the  perimeter  of  a  circle  halfway  between  the  bases. 

QUERY.  How  do  you  interpret  the  formulae  for  the  volume  and  the 
lateral  area  of  a  frustum  of  a  cone  when  r  =  0;  when  r  =  rt? 

EXERCISES 

143.  A  right  circular  cone  is  14  inches  high  and  has  a  radius 
of  5  inches.   Find  its  lateral  area. 

144.  A  frustum  of  a  right  circular  cone  is  10  inches  high,  and 
its  radii  are  4  and  6  respectively.    Find  the  total  area. 


416 


SOLID  GEOMETRY 


145.  A  tent  of  radius  16  feet  and  14  feet  high  is  in  the  form  of 
a  cylinder  surmounted  by  a  cone.    The  height  of  the  conical  part 
is  8  feet.   Find  the  number  of  square  yards  in  the  surface  of  the  tent. 

146.  Find  the  lateral  area  and  the  total  area  of  a  right  circular 
cone  whose  volume  is  24  TT  cubic  feet  and  whose  radius  is  3  feet. 

147.  Prove  that  the  areas  of  circular  sections  of  a  cone  made 
by  parallel  planes  are  proportional  to  the  squares  of  the  distances 
of  their  planes  from  the  vertex. 

148.  The  radii  of  circular  sections  of  a  cone  made  by  parallel 
planes  are  proportional  to  their  distances  from  the  vertex. 

Theorem  27 

560.  A  right  circular  cone  is  generated  by  the  revolution 
of  a  right  triangle  about  one  of  its  sides  as  an  axis. 


Given  any  right  triangle  XOA  revolving  about  the  side  OX  as 
an  axis  and  generating  the  figure  OAK. 

To  prove  that    OAK  is  a  right  circular  cone. 
Proof.  XA  generates  a  plane  _L  to  XO  at  X. 

A  generates  a  circle  with  center  at  A'. 

OA  generates  a  conical  surface. 
Therefore  XOA  generates  a  right  circular  cone. 


Why? 

Why? 
§535 
§542 


561.  Cone  of  revolution.    In  accordance  with  Theorem  27, 
a  right  circular  cone  is  frequently  called  a  cone  of  revolution. 


BOOK  VII 


417 


562.  Similar  cones.    Two  cones  of   revolution   which  are 
generated  by  similar  right  trian- 

gles with  corresponding  sides  as 
axes  are  similar  cones  of  revolution. 

QUERY  1.  Would  two  cones  of  revo- 
lution ever  be  similar  if  they  were 
formed  by  the  revolution  of  similar 
triangles  about  sides  that  were  not 
corresponding  ? 

QUERY  2.  Would  a  cone  of  revolution  be  formed  by  the  revolution 
of  a  triangle  about  its  altitude  as  an  axis? 

Theorem  28 

563.  If  two  cones  of  revolution  are  similar, 

(1)  their  volumes  are  proportional  to  the  cubes  of  their 
altitudes,  of  their  radii,  or  of  their  slant  heights  ; 

(2)  the  lateral  areas  or  the  total  areas  are  proportional 
to  the  squares  of  their  altitudes,  of  their  radii,  or  of  their 
slant  heights. 

Let  F,  S,  T,  h,  r,  and  I  stand  for  volume,  lateral  area,  total 
area,  altitude,  radius,  and  slant  height,  respectively.    Then 


*     ___________ 

*  *' 


The  proof,  which  is  left  to  the  stu- 
dent, is  similar  to  that  of  Theorem  10. 

564.  Tangent  plane  to  a  cone.  A  tan- 
gent plane  to  a  cone  is  one  which  meets 
the  cone  only  along  an  element. 

A  tangent  plane  to  a  cone  is  deter- 
mined by  a  tangent  to  the  base  of  the 
cone  and  the  element  drawn  to  the 
point  of  contact. 


418 


SOLID  GEOMETRY 


565.  Circumscribed  pyramid.  A  pyramid  is  said  to  be  cir- 
cumscribed about  a  cone  when  its  base  is  circumscribed  about 
that  of  the  cone  and  its  lateral  faces 
are  tangent  to  the  cone. 

QUERY  1.  What  point  do  all  the  tangent 
planes  to  a  cone  have  in  common? 

QUERY  2.  How  many  planes  tangent  to  a 
given  circular  cone  can  be  passed  through 
a  given  exterior  point? 

QUERY  3.  Can  a  right  triangle  be  found 
which  will  generate  any  given  right  circular 
cone? 

QUERY  4.  What  does  a  right  triangle  generate  when  revolved  about 
its  hypotenuse? 

QUERY  5.  Describe  the  solid  generated  by  the  revolution  of  an 
obtuse  triangle  about  one  side  of  the  obtuse  angle. 

QUERY  6.  Describe  the  solid  generated  by  an  isosceles  trapezoid 
when  revolved  about  (1)  the  line  joining  the  mid-points  of  the  parallel 
sides,  (2)  the  longer  parallel  side,  (3)  the  shorter  parallel  side. 

QUERY  7.  What  does  a  square  generate  when  revolved  about  a 
diagonal  ? 

QUERY  8.  What  does  a  rectangle  generate  when  revolved  about  an 
axis  parallel  to  one  of  the  sides  of  the  rectangle  and  in  its  plane  but 
lying  entirely  outside  the  rectangle? 

QUERY  9.  Similar  polyhedrons  have  not  yet  been  denned  in  this  text. 
But  from  analogy  with  the  ratio  of  the  areas  of  similar  polygons,  and 
with  the  results  of  Theorems  10  and  28  in  mind,  what  do  you  think  the 
ratio  of  the  volumes  of  similar  polyhedrons  probably  is  ? 

EXERCISES 

In  Exercises  14§-157,  .which  refer  to  cones  of  revolution, 
h  represents  the  altitude,  r  the  radius  of  the  base,  I  the  slant 
height,  S  the  lateral  area,  T  the  total  area,  and  V  the  volume. 

149.  Given  r  =  5J,  h  =  8f    Find  V. 

150.  Given  r  =  6,  I  =  5.    Find  S. 

151.  Given  r  =  5,  I  =  8.    Find  V. 


BOOK  VII 


419 


152.  Given  I  =  6,  S  =  132.    Find  T. 

153.  Given  T  =  55,  r  =  3.    Find  S. 

154.  Given  V=  110,  r  =  4.    Find  S. 

155.  Given  T  =  2  S,  r  =  5.    Find  V. 

156.  Given  F  =  S,  I  =  4.    Find  V. 

157.  Given  J  =  2  r,  /*,  =  2  V3.    Find  r. 

158.  The  sides  of  a  right  triangle  are  6,  8,  and  10  inches.    Find 

(1)  the  volume,  (2)  the  lateral  area,  and  (3)  the  total  area  of  the 
cone  generated  by  revolving  the  triangle  about  its  shortest  side. 

159.  The  triangle  of  Exercise  158  is  revolved  about  the  side  8. 
Find  (1)  the  volume,  (2)  the  lateral  area,  and  (3)  the  total  area, 
and  compare  these  values  with  the  results  of  Exercise  158. 

160.  Find  (1)  the  volume  and  (2)  the  area  of  the  solid  formed 
by  the  rotation  of  an  equilateral  triangle  about  one  of  its  altitudes. 

161.  The  parallel  sides  of  an  isosceles  trapezoid  are  12  and  18 
inches  respectively.   The  altitude  is  4  inches.  Find  (1)  the  volume, 

(2)  the  area  of  the  solid  obtained  by  rotating  it  about  the  longer 
of  the  parallel  sides. 

162.  What  is  the  area  of  the  largest  section  which  can  be  made 
by  a  plane  through  the  vertex  of  a  right  circular  cone  whose  alti- 
tude is  10  inches  and  whose  radius  is  6  inches. 

163.  Prove  that  a  cone  circumscribed  about  a  regular  pyramid  is 
a  cone  of  revolution. 

POLYHEDRONS 

566.  Polyhedral  angle.    The  figure  formed  at  the  vertex  of 
a  pyramid  by  one  nappe  of  a  pyramidal 
surface  (see  §  513)  is  called  a  polyhedral 
angle. 

The  principal  distinction  between  the 
polyhedral  angle  and  the  pyramidal  sur- 
face lies  in  the  fact  that  in  the  former  we 
fix  the  attention  on  the  portion  of  the  figure 
near  the  vertex. 


420 


SOLID  GEOMETRY 


The  terms  face,  edge,  vertex^  and  dihedral  angle  are  applied 
to  the  parts  of  the  polyhedral  angle  in  the  same  sense  as 
they  are  applied  to  the  parts  of  a  pyram- 
idal  surface. 

567.  The  angle  between  two  edges  of  a 
polyhedral  angle  is  called  a  face  angle. 

568.  Trihedral  angle.  A  polyhedral  angle 
which   has   only  three   faces  is  called   a 
trihedral  angle. 

Theorem  29 

569.  If  two   trihedral   angles   have   their  face   angles 
equal  and  arranged  in  the  same  order,  ike  corresponding 
dihedral  angles  are  equal. 


Given  any  trihedral  angles  0  and  Of,  in  which  the  face  angle 
XOY  equals  X'0'Y>,  YOZ  equals  Y'O'Z',  ZOX  equals  Z'O'X'. 

To  prove  the  dihedrals  OX=0'X',  OY=0'Y',  OZ=0'Z'. 
Proof.    Lay  off  OA,  OB,  OC,  equal  respectively  to  O'A ',  O'B',  O'C'. 
Pass  planes  through  ABC  and  A'B'C'. 


BOOK  VII  421 

Lay  off  A K  equal  to  A  'K',  and  pass  KLM  and  K'L'M'  J_  respec- 
tively to  OA  and  O'A'. 

AK  is  _L  to  KL  and  KM,  while  A  'K'  is±toK'L'  and  K'M'.     §  414 

Hence  the  angles  LKM  and  L'K'M1  are  plane  A  of  dihedrals 
OA  and  O'A1  respectively.  §  432 

We  shall  prove  the  plane  angles  equal  by  proving  A  LKM 
congruent  to  A  L'K'M'. 

A  A  OB  is  congruent  to  AA'O'B'.  §  25 

Hence                            Z  KA  L  =  ^K'A'L '.  Why  ? 

Therefore    right  A. 4  KL  is  congruent  to  right  A  AWL'.    Why? 

Similarly,       A  A  KM  is  congruent  to  AA'K'M'. 

Hence                   KL  =  K'L'  and  KM  =  K 'M'.  §  27 

It  remains  to  show  that    LM=  L'M\ 

Now                                        AB  =  A'B'.  Why? 

Similarly,  BC  =  B'C'  and  CV1  =  CU'. 

Hence  A  ABC  is  congruent  to  AA'B'C'  Why? 

and  ZC4£=ZC"4'£'.  Why? 

Furthermore,       A  L  =  A  'L'  and  A  M  =  A  'M'.  Why  ? 

Therefore  AMAL  is  congruent  to  AMM'Z'  §  27 

and  LM=L'M'.  Why? 

Hence  ALKM  is  congruent  to  AL'K'M'  Why? 

and  ^  LKM  =  AL'K'M!.  Why? 

Therefore             dihedral  ,40=  dihedral  ^  '0'.  Why  ? 

After  similar  constructions  on  the  other  edges  of  the  trihedral 
it  can  be  proved  by  the  method  just  employed  that 
dihedrals  OB  and  OC  =  dihedrals  O'B'  and  0'C"  respectively. 

Hence  the  three  dihedrals  of  one  trihedral  angle  are  equal 
respectively  to  the  three  corresponding  dihedrals  of  the  other. 


422  SOLID  GEOMETRY 

EXERCISES 

164.  If  the  face  angles  of  two  trihedrals  are  equal  each  to  each 
and  are  arranged  in  the  same  order,  the  trihedrals  are  congruent. 

165.  If  two  trihedral  angles  have  a  face  angle  of  one  equal  to 
a  face  angle  of  the  other,  and  the  corresponding  adjacent  dihedrals 
equal,  prove  by  superposition  that  the  trihedrals  are  congruent. 

166.  If  the  edges  of  two  trihedral  angles  are  parallel  each  to 
each  and  extend  in  the  same  directions  from  the  vertices,  the 
dihedrals  are  congruent. 

QUERY  1.  Can  each  of  three  planes  meet  the  other  two  without 
forming  a  trihedral  angle  ? 

QUERY  2.  How  small  may  the  sum  of  the  face  angles  of  a  trihedral 
angle  be?  How  large? 

QUERY  3.  How  does  a  trihedral  angle  look  if  the  sum  of  its  dihedral 
angles  is  nearly  equal  to  six  right  angles  ?  How  if  the  sum  of  its  dihe- 
dral angles  is  nearly  as  small  as  two  right  angles  ? 

QUERY  4.  Referring  to  the  propositions  proved  in  Plane  Geometry, 
it  appears  that  Theorem  29  and  Exercise  165  above  correspond  to  §  33 
and  §  82,  respectively,  of  the  Plane  Geometry  if  we  relate  the  face 
angles  and  dihedrals  of  the  trihedral  to  the  sides  and  angles  of  the 
triangle  respectively. 

Read  the  list  of  theorems  proved  in  Plane  Geometry  regarding  the 
triangle,  and  state  those  to  which  you  think  a  corresponding  theorem 
for  trihedral  angles  seems  likely  to  be  true.  Consider,  for  example,  §§  25, 
29,  and  66. 

QUERY  5.  If  isosceles  trihedral  angles  are  defined  as  those,  two  of 
whose  face  angles  are  equal,  study  the  correspondence  between  the 
theorems  on  isosceles  triangles  and  isosceles  trihedrals. 

NOTE.  In  plane  geometry,  if  two  triangles  have  three  angles  of  one 
equal  respectively  to  three  angles  of  the  other,  the  triangles  are  not 
congruent  but  merely  similar.  The  corresponding  problem  relating  to 
trihedral  angles  would  be  that  of  finding  the  relation  between  two 
trihedrals  whose  dihedrals  are  respectively  equal.  Asa  matter  of  fact 
such  trihedrals  are  congruent,  not  similar,  as  one  would  expect  from 
plane  geometry.  This  fact  cannot  be  proved  simply  until  a  little  later, 
in  connection  with  our  study  of  the  sphere. 


BOOK  VII  423 

Theorem  30 

570.   The  sum  of  two  face  angles  of  a  trihedral  angle 
is  greater  than  the  third. 

O 


Given  any  trihedral  angle  0-XYZ. 
To  prove  that    Z.ZOY+  ^XOZ 

Proof.    There  is  no  necessity  for  proof  if  either  XOZ  or  ZOY 
is  greater  than  or  equal  to  XOY. 

In  the  face  XOY  draw  OK,  making  the  /.XOK  =  /.XOZ. 
On  OZ  lay  off  OC  =  OK. 

Through  K  and  C  pass  a  plane  which  does  not  contain  0,  as  ABC. 
In  A  0,4  #  and  OAC, 

AO  =  AO,  OK=  OC,  ^AOK  =  Z.AOC.  Why? 

Hence  A  A  OK  is  congruent  to  AAOC.  Why? 

Therefore  AK  =  AC. 

in  AABC,          AC  +  BOAK  +  KB.  §146 

Subtracting  the  equals  AC  and  AK,  we  have  BC  >  KB.     §  139 
In  the  A  0KB  and  OBC,  OK=  OC,  OB  =  OB,  CB  >  KB. 
Therefore  Z.BOC  >  Z.KOE.  §  151 

Hence,  adding  to  this  inequality  the  equal  angles  A  OC  and  A  OK, 
we  have         ^LAOC  +  Z.BOC  >Z.AOK  +  ^KOB.  §139 


571.  Corollary.    Any  face  angle  of  a  polyhedral  angle  is  less 
than  the  sum  of  the  remaining  face  angles, 


424  SOLID  GEOMETRY 

Theorem  31 

572.   The  sum  of  the  face  angles  of  any  convex  poly- 
hedral angle  is  less  than  four  right  angles. 


X 

\Z 

Given  any  polyhedral  angle  P with  face  angles  XPY,  YPZ,  etc. 
To  prove  that          XPY  +  YPZ+  .  .  .  <  4  right  angles. 

Proof.    Pass  a  plane  forming  the  section  A  ROD. 
Let  the  sum  of  the  face  angles  of  P  be  denoted  by  F. 
Let  the  sum  of  the  remaining  angles  in  the  lateral  faces,  as 
A  HP,  PRO,  etc.,  be  denoted  by  L. 

Let  the  sum  of  the  angles  of  the  polygon  ARCD  be  denoted  by  5. 
Then  if  the  polyhedral  angle  has  n  faces, 

F+L  =  2ni-t.A.  (1)  §66 

Also  S  =  (2  n  -  4)  rt.  A.  (2)         §  125 

Now  ARP  +  PRC.>ARC.  §570 

Since  a  similar  inequality  holds  at  each  vertex  of  the  polygon 
A-D,  we  have  L>g  §  ^ 

Hence  L  >  (2  n  -  4)  rt.  A.  (3) 

Subtracting  (3)  from  (1), 

F  <  4  right  angles.  §  142 

573.  Regular  polyhedron.  A  convex  polyhedron  is  said  to 
be  regular  if  its  faces  are  all  congruent  regular  polygons  and 
its  polyhedral  angles  are  all  congruent. 


BOOK  VII 
Theorem  32 


425 


574.  No  more  than  five  regular  polyhedrons  are  possible. 

Proof.  Each  of  the  polyhedral  angles  of  a  regular  convex 
polyhedron  is  included  by  three  or  more  faces  which  are  regular 
polygons.  Consider  in  turn  the  possibilities  when  the  faces  are 
(1)  equilateral  triangles,  (2)  squares,  (3)  pentagons,  (4)  hexagons. 


(1)  Each  angle  of  an  equilateral  triangle  contains  60°.    Hence 
there  may  be  convex  polyhedral  angles  with  three,  four,  or  five 
such  faces,  but  not  with  six  or  more.  §  572 

Hence  there  can  be  no  more  than  three  regular  polyhedrons 
with  triangular  faces. 

(2)  Each  angle  of  a  square  contains  90°. 
There  may  be  convex  polyhedral  angles  with 
three  square  faces,  but  not  with  four  or  more. 

Hence  there  can  be  no  more  than  one  regular 
polyhedron  with  square  faces. 

(3)  Each  angle  of  a  regular  pentagon  con- 
tains 108°.    Hence  there  may  be  convex  poly- 
hedral angles  with  three  pentagons  for  faces 
but  not  with  four  or  more.  Why  ? 

Hence  there  can  be  no  more  than  one  regular 
polyhedron  with  pentagonal  faces. 

(4)  Each  angle  of  a  hexagon  contains  120°. 

Hence  there  can  be  no  regular  polyhedrons  with  hexagonal  sides. 

In  the  case  of  polygons  of  more  than  six  sides,  no  polyhedral 

angle  can  be  formed.  Why  ? 

Therefore  no  more  than  five  regular  polyhedrons  are  possible. 


426 


SOLID  GEOMETRY 


575.  The  five  regular  bodies.  It  does  not  follow  from  the 
foregoing  theorem  that  there  are  necessarily  five  regular  poly- 
hedrons, but  merely  that  this  is  the  maximum  possible  num- 
ber. In  order  to  show  the  existence  of  polyhedrons  of.  the 
various  kinds  it  is  necessary  to  prove  the  possibility  of  con- 
structing each  type.  As  a  matter  of  fact,  each  of  the  five 
cases  referred  to  in  Theorem  32  does  actually  correspond  to 
a  regular  polyhedron,  as  shown  in  the  following  diagrams: 


TETRAHEDKON    HEXAHEDRON 

(Cube) 


OCTAHEDRON     DODECAHEDRON    ICOSAHEDRON 


NOTE.  The  names  of  the  regular  polyhedrons  are  derived  from  the 
Greek  words  for  four,  six,  eight,  twelve,  and  twenty  respectively,  which 
reference  to  the  diagrams  will  show  to  be  the  numbers  of  the  faces  of 
the  various  polyhedrons. 

Construction  1 
576.   Construct  a  regular  tetrahedron. 


Construction.    Construct   an    equilateral   A  ABC,  and  find   its 
center,  P. 

At  P  construct  a  line  _L  to  the  plane  of  ABC.         §  420 


BOOK  VII 


427 


With  a  vertex  of  ABC  as  center,  and  a  side  of  ABC  as  radius, 
determine  0,  in  this  J_,  so  that  AO  =  AB. 

Draw  OA,  OB,  OC. 


Then  OA  B  C  is  a  regular  tetrahedron. 

Proof.  OA  =  OB  =  OC. 

But  AB  =  AO. 

Hence  A  OB  is  an  equilateral  A. 

Similarly,  the  other  faces  are  equilateral  A. 

Therefore  all  the  trihedral  A  are  congruent. 

Hence  the  tetrahedron  is  regular. 

Construction  2 

577.   Construct  a  regular  hexahedron,  or  cube. 
M  L 


§425 
Const. 


§569 
§573 


.XT 


Construction.    Construct  a  square,  A  BCD,  and  at  each  of  the 
four  vertices  erect  lines  _L  to  the  plane  of  the  square.  §  420 

Lay  off  along  these  Js  AP  =  BK  =  CL  =  DM,  each  =  AB. 

Pass  planes  PKB,  KLC,  LMD,  and  MPA.  §  385 

Through  P  pass  a  plane  II  to  AC.  §  408 

The  resulting  figure  is  a  cube, 
Proof  is  left  to  the  student, 


428 


SOLID  GEOMETRY 


Construction  3 
578.   Construct  a  regular  octahedron. 


The  Construction  and  Proof  are  left  to  the  student. 

HINT.  Construct  two  square  regular  pyramids  having  the  same  base, 
with  vertices  on  opposite  sides  of  the  base  and  with  lateral  edges  equal 
to  the  sides  of  the  base. 

NOTE.  The  constructions  of  the  dodecahedron  and  icosahedron  are 
much  more  complicated  and  will  not  be  given  here. 


V_A_/     kx> 


Models  of  all  the  regular  solids  can  easily  be  made  by  cutting  card- 
boards in  the  forms  of  the  above  diagrams,  then  c,utting  half  through 
the  cardboard  along  the  dotted  lines,  folding  along  the  half-cuts,  and 
closing  the  model  by  pasting  strips  of  paper  along  the  open  edges. 


BOOK  VII  429 

EXERCISES 

167.  Prove  that  any  two  pairs  of  opposite  vertices  of  a  regular 
octahedron  are  the  vertices  of  a  square. 

168.  The  side  of  a  cube  is  6  inches.    Find  the  side  of  a  cube 
of  twice  the  volume. 

NOTE.  To  Hippocrates  of  Chios  (about  430  B.C.)  is  due  the  proof  that 
the  solution  of  the  problem  of  duplicating  the  cube  can  be  reduced  to 
the  finding  of  two  mean  proportionals  between  two  given  lines,  of  which 
one  is  the  side  of  the  given  cube  and  the  other  is  twice  that  side. 

If  x  and  y  are  two  mean  proportionals  between  a  and  2  a, 
wehave  a:x  =  x:y  =  y:2a. 

Then  x2  =  ay  and  y2  =  2  ax. 

Squaring  the  first,  a;4  =  a'2?/2. 

Substituting  value  of  ?/2  from  the  second, 

x*  =  a2  •  2  ax  =  2  a3x. 
Hence  x3  =  2  a3. 

That  is,  the  volume  of  the  cube  of  edge  x  will  be  double  that  of  a  cube 
with  edge  a. 

The  geometric  procedure  for  the  duplication  of  the  cube  has  been 
carried  out  in  a  variety  of  ways,  usually  by  finding  the  intersection  of 
various  curves  such  as  parabolas  or  hyperbolas.  Plato  (400  B.C.)  is  said 
to  have  solved  the  problem  by  means  of  a  mechanical  device,  but  to 
have  rejected  the  method  as  not  being  geometric. 

169.  One  edge  of  a  regular  octahedron  is  8  inches.    Find  the 
volume. 

170.  A  cube  and  a  regular  tetrahedron  have  the  same  edge. 
What  is  the  ratio  of  their  volumes  ? 

HINT.    See  Exercise  126. 

171.  The  base  of  a  regular  hexagonal  pyramid  is  66  square 
inches,  and  its  altitude  is  10  inches.     A  plane  is  passed  parallel 
to  the  base  5  inches  from  it.    Find  the  ratio  of  the  volume  of  the 
original  pyramid  to  that  of  the  one  cut  off  by  this  plane. 


430 


SOLID  GEOMETRY 
Theorem  33 


579.  If  two  tetrahedrons  have  a  trihedral  angle  of  one 
congruent  to  a  trihedral  angle  of  the  other,  their  volumes 
are  in  the  same  ratio  as  the  products  of  the  edges  of  these 
trihedral  angles. 


Given  0-ABC  and  P-KLM,  with  trihedrals  0  and  P  congruent. 

,   ,  volume   0-ABC        OA  •  OB  •  OC 

To  prove  that  —      =  —  — 

volume  P-KLM     PK  •  PL  •  PM 

Proof.    Apply  the  trihedral   0  to  P  so  that  the  tetrahedron 
O-ABC  takes  the  position  P-RST. 

Let  TX  and  MY  be  the  altitudes  of  the  two  tetrahedrons. 

volume  P-RST        PRS  •  TX       PRS     TX 

—  •    (1)  §553 


Now 


volume  P-KLM      PKL -MY      PKL    MY 

TX  is  II  to  MY,  §  427 

and  the  plane  determined  by  TX  and  MY  contains  PM  and  hence 
the  APTX  and  PMY. 

Therefore  APTX  is  similar  to  A  PMY.  §  271 

TX       PT 


_, 
Therefore 

In  the  APRS  and  PKL,  we  have 

PRS  _  PR  .  PS  _  PR 
PKL  ~  PK  •  PL  "~  'PK 


PS 
PL 


(2)  Why? 


(3)     §  325 


BOOK  VII 


431 


or 


Hence,  substituting  (3)  and  (2)  in  (1), 

volume   P-RST  _  PR    PS    PT 
volume  P-KLM      PK    PL    PM* 
volume   0-ABC  _  OA  •  OB  .  PC 
volume  P-KLM  ~~  1'K  -  PL  -  PM 


580.  Symmetric  trihedrals.  If  two  trihedral  angles  have 
their  parts  equal  each  to  each,  but  arranged  in  the  opposite 
order,  they  are  called 
symmetric. 

For  example,  the 
trihedral  angles  in 
the  two  nappes  of 
a  triangular  pyram- 
idal surface  are  sym- 
metric. For  if  one 
thinks  of  oneself  as 

stationed  at  the  vertex  and  looking  first  in  the  direction 
of  the  faces  of  one  angle  and  then  in  that  of  the  faces  of 
the  other,  the  equal  dihedral  angles  OA,  OB,  0(7  in  the  diagram 
follow  each  other  in  clockwise  order  on  the 
lower  nappe  and  in  counterclockwise  order 
on  the  upper  nappe. 

QUERY  1.  Can  two  triangles  whose  parts  are  equal 
but  arranged  in  opposite  order  be  superposed  so  as  to 
coincide  throughout  ? 

QUERY  2.  Can  two  symmetric  trihedral  angles  be 
superposed  so  as  to  coincide  throughout? 

QUERY  3.  Are  the  dihedral  angles  of  a  regular 
polyhedron  all  equal  ? 

EXERCISE  172.  If  two  trihedral  angles  have  their  edges  paral- 
lel, and  all  the  edges  of  one  trihedral  extending  in  directions  from 
the  vertex  opposite  to  the  corresponding  edges  of  the  other  tri- 
hedral, the  trihedrals  are  symmetric. 


432 


SOLID  GEOMETRY 


581.  Similar  polyhedrons.    Two  polyhedrons  are  similar  if 
their  faces  are  similar,  each  to  each,  and  similarly  placed,  and 
if  their  corresponding  polyhedral  angles  are  congruent. 

It  should  be  observed  that  similar  solids  have  the  same  shape 
but  different  sizes. 

Theorem  34 

582.  The  ratio  of  any  two  corresponding  edges  of  two 
similar  polyhedrons  is  equal  to  the  ratio  of  any  other 
corresponding  pair. 


Proof  is  left  to  the  student. 

QUERY  1.    Are  all  regular  polyhedrons  with  the  same  number  of 
faces  similar? 

QUERY  2.    Are  all  rectangular  solids  similar? 

EXERCISES 

173.  The  areas  of  two  corresponding  faces  of  two  similar  poly- 
hedrons are  in  the  same  ratio  as  the  squares  of  any  two  corre- 
sponding edges. 

174.  Two  tetrahedrons  are  similar  if  three  faces  of  one  are 
similar  and  similarly  placed  to  three  faces  of  the  other. 

175.  Two  tetrahedrons  are  similar  if  a  dihedral  in  one  is  equal 
to  a  dihedral  in  the  other,  and  if  the  faces  including  the  dihedral 
are  similar  and  similarly  placed. 

176.  If  a  plane  is  passed  parallel  to  the  base  of  a  pyramid,  the 
pyramid  cut  off  is  similar  to  the  original  one. 


BOOK  VII  433 

Theorem  35 


583.    The  volumes  of  two  similar  tetrahedrons  are  in  the 
same  ratio  as  the  cubes  of  two  corresponding  edges. 


AkV>C 
B 


Given  two  similar  tetrahedrons  0-ABC  and  0'-A'£rCf,  with 
volumes  Fand  V  respectively. 

V      O43 


,  4 
To  prove  that 

V      Q'A'S 
Proof.         Trihedral  0  is  congruent  to  trihedral  O'.  §  581 

V          PA-OB-  PC  S._Q 

T'^O'A'.O'B'.O'C' 
OA      OB      PC 
~  O'A1'  O'B1'  O'C"' 

The  A  OAB  and  O'A'B',  OA  C  and  O'A  'C',  etc.  are  similar.    §  581 
OA         OB        PC 


Substituting  (2)  in  (1),  we  obtain 


584.  Assumption.  Any  two  similar  polyhedrons  can  be 
divided  into  the  same  number  of  tetrahedrons,  similar  each 
to  each  and  similarly  placed. 


434  SOLID  GEOMETRY 

The  foregoing  statement,  a  proof  of  which  is  rather  involved, 
appears  evident  if  we  consider  two  corresponding  points  (one  in 
the  interior  of  each  polyhedron)  and  with  these  points  as  ver-tices 
construct  pyramids  having  as  their  bases  the  various  faces  of  the 
polyhedrons.  The  pyramids  with  similar  faces  for  bases  are  sim- 
ilar, and  hence  the  tetrahedrons  into  which  they  may  be  divided 
are  similar  and  similarly  placed. 


Theorem  36 

585.  The  volumes  of  two  similar  polyhedrons  are  in  the 
same  ratio  as  the  cubes  of  any  two  corresponding  edges. 

Given  two  similar  polyhedrons  P  and  Pr,  with  volumes  V 
and  V1  and  corresponding  edges  E  and  E'  respectively. 

V      E* 
To  prove  that  -  -=  =  —  . 

Proof.  Divide  P  and  P'  into  tetrahedrons  which  are  similar 
each  to  each  and  similarly  placed.  §  584 

Let  the  volumes  of  the  similar  tetrahedrons  be  denoted  by  Tly  T[; 
Ti,  Ti\  Tz,  TZ,  etc.,  and  let  corresponding  edges  of  th'ese  tetra- 
hedrons be  denoted  by  Elt  E[-,  E^  E^  EB,  E'3J  etc. 

Then  =        ;        =  |4'  etc.  §  583 

2  &t 

•••=•  §582 


Therefore  =        =  .  .  .  =  ~  Why? 


Hence 


T2+    Ti 

V        1 


or 


BOOK  VII  485 

586.  Similar  figures.    We  can  now  summarize  the  whole 
doctrine  of  similar  figures  by  the  following  statements: 

Similar  plane  figures  or  similar  surfaces  are  in  the  same 
ratio  as  the  squares  of  any  two  corresponding  lines. 

Similar  solids  are  in  the  same  ratio  as  the  cubes  of  any  two 
corresponding  lines. 

In  similar  figures  of  any  kind,  pairs  of  corresponding  lines 
are  in  the  same  ratio. 

EXERCISES 

177.  If  one  edge  of  a  polyhedron  is  6  inches,  what  is  the 
length  of  the  corresponding  edge  of  a  polyhedron  of  twice  the 
volume  ? 

178.  If  the  base  of  one  pyramid  has  nine  times  the  area  of  the 
base  of  a  similar  pyramid,  what  is  the  ratio  of  their  volumes  ? 

179.  If  a  cube  has  an  edge  8  inches  long,  what  is  the  diagonal 
of  a  cube  of  four  times  the  volume  ? 

180.  A  regular  tetrahedron  has  a  volume  of  27  cubic  inches. 
What  is  the  volume  of  a  similar  tetrahedron  whose  edges  are  each 
half  as  long  ? 

181.  If  the  strength  of  two  steel  wires  varies  directly  as  their 
cross  section,  what  is  the  ratio  of  two  weights  that  can  just  be 
supported  by  wires,  one  of  which  is  three  times  as  great  in 
diameter  as  the  other  ? 

587.  Prismatoids.   A  prismatoid    is    a   polyhedron    all    of 
whose  vertices  lie  in  two  parallel  planes. 

The  lateral  faces  of  prismatoids  are  either 
triangles  or  quadrilaterals.  Pyramids,  prisms, 
and  frustums  of  pyramids  are  special  cases  of 
prismatoids. 

The  terms  base,  altitude,  etc.  are  defined  for 
the  prismatoid  similarly  to  the  corresponding  terms  as  applied 
to  the  prism. 


436 


SOLID  GEOMETRY 
Theorem  37 


588.  The  volume  of  a  prismatoid  equals  the  product  of 
one  sixth  the  altitude  by  the  sum  of  the  upper  base,  the 
lower  base,  and  four  times  the  mid-section. 

A C  JC 


Given  the  prismatoid  ABCDEFG  in  which  the  upper  base,  the 
lower  base,  the  mid-section,  and  the  altitude  are  denoted  by 
bj  B,  M,  and  h  respectively. 

h 
To  prove  that  V=  -  (6  +  ^  +  4  Jf). 

Join  any  point  in  the  mid-section,  as  0}  with  the  various  vertices 
of  the  mid-section  and  of  both  bases  by  the  lines  OB,  OR,  OF, 
etc.  The  planes  passed  through  these  lines  divide  the  figure  into 
pyramids  with  a  common  vertex  at  0,  like  0-ABC,  0-DEFG,  which 
have  as  their  bases  b  and  .B  respectively,  and  0-BCF,  whose  base 
is  one  of  the  triangular  lateral  faces  of  the  prismatoid. 

The  volumes  of  these  pyramids  are  found  separately. 

(a)  Volume.  0-ABC  =  J  -  \  - 1  =  ~- 

O      2i  D 

(/>)  Volume  0-DEFG  =  -  •  | .  B  =  ^  - 
o    2i  D 

(c)  Volume  O—BCF  is  divided  into  two  parts  by  the  mid-section. 
Since  R S  joins  the  mid-points  of  the  sides  of  BCF, 

AFRS=±AFBC,  and  volume  0-FRS  =  J  0-FBC.   §  553 


BOOK  VII  437 

}  ORS  =  p  (ORS). 


But  0-FRS  =  F-ORS  =  -(-h 
o 

Therefore  volume  0-FBC  =  4 (0-FRS)  =  ±  •  ^  •  ORsJ^h  .  ORS. 

D  O 

In  a  similar  manner  the  volume  of  the  other  portions  of  the 
figure  not  considered  under  (a)  and  (b)  can  be  shown  to  equal  -§  h 
times  the  area  of  the  mid-section  included  in  it. 

Hence  by  adding  together  the  portions  of  the  prismatoid  not 
considered  in  (a)  and  (b)  we  obtain  as  their  volume  §  h  •  M. 

Adding  (a),  (b),  and  this  last  result, 

NOTE.  It  can  be  proved  by  the  integral  calculus  that  the  foregoing 
formula  is  valid  for  a  much  more  extensive  class  of  solids  than  prism  a- 
toids.  It  is  usually  called  the  Prismoidal  Formula.  Among  these  solids 
are  included  cones,  cylinders,  and  spheres.  In  fact,  the  class  of  solids 
whose  volume  can  be  found  by  means  of  (1)  is  so  extensive  that  the  for- 
mula finds  wide  practical  application.  The  formula  was  first  stated  by 
Thomas  Simpson  (1710-1701),  who  also  gave  a  list  of  solids  for  which  it 
gives  accurate  results.  It  may,  for  example,  be  assumed  that  a  pile  of 
earth 'or  sand  or  the  excavation  for  a  foundation  or  a  railway  cut  is  so 
nearly  in  the  form  of  such 
a  solid  that  its  volume 
may  be  computed  by  the 
use  of  (1)  without  appreci- 
able error. 

For  example,  suppose 
an    excavation   100    feet 

square  at  the  top  and  10  feet  deep  is  to  be  made  in  sand  so  that  the 
sides  of  the  excavation  form  an  angle,  of  45°  with  the  horizontal. 
The  bottom  of  the  excavation  is  then  a  square  80  feet  on  a  side,  and 
the  area  of  the  section  halfway  down  is  a  square  90  feet  on  a  side. 

Hence  V=  ^  (802  +  1002  +  4  x  902) 

D 

=  IPX  48,800  =  81;,33c,,i,icfeet. 


438  SOLID  GEOMETRY 

EXERCISES 

182.  Show  that  the  expression  for  the  volume  of  a  pyramid  is 
a  special  case  of  (1).    In  the  case  of  the  pyramid,  b  =  o,  B  —  B, 

M=  i  B.    Hence  (1)  becomes  F=  |  (0  +  B  +  B)  =  ^j-- 

183.  Show  that  the  expression  for  the  volume  of  the  prism  is 
given  by  the  Prismoidal  Formula. 

184.  Show  that  the  volume  of  a  cone,  the  volume  of  the  frustum 
of  a  cone,  and  the  volume  of  the  frustum  of  a  pyramid  are  each 
given  by  the  Prismoidal  Formula. 

185.  An  irregular  pile  of  earth  is  15  feet  high  and  covers  500 
square  feet.    Its  mid-section  and  level  top  are  estimated  to  con- 
tain 400  and  200  square  feet  respectively.    Find  the  cost  of  re- 
moving it  at  50  cents  per  load,  if  the  truck  measures  3x4x9  feet. 

186.  A  circular  pile  of  sand  which  stands  at  rest  at  an  angle 
of  45°  is  6  feet  high.    How  many  cubic  feet  does  it  contain  ? 

187.  An  excavation  must  be  carried  20'  deep  in  earth  which  at 
rest  stands  60°  to  the  horizontal.    The  bottom  must  be  a  square 
40'  x  40'.    (1)  How  large  are  the  top  and  the  mid-section  of  the 
excavation  ?   (2)  How  many  cubic  feet  of  earth  must  be  removed  ? 

REVIEW  EXERCISES 

188.  A  cubic  foot  of  water  weighs  about  62  pounds.    What 
must  be  the  side  of  a  square  refrigerator  pan  6  inches  'high  in 
order  to  hold  the  water  from  50  pounds  of  ice  ?    How  many  square 
inches  of  sheet  tin  are  required  to  make  the  pan,  allowing  half  an 
inch  along  each  seam  for  folding  ? 

189.  A   trough   is    formed    by 
nailing    together,    edge    to    edge, 
two  boards  12  feet  in  length,  so 

that  the  right  section  is  a  right  angle.  If  15  gallons  of  water  is 
poured  into  the  trough  and  it  is  held  level  so  that  a  right  section 
of  the  water  is  an  isosceles  right  triangle,  how  deep  is  the  water  ? 
(231  cubic  inches  =  1  gallon.) 


BOOK  VII  439 

190.  Two  water  tanks  in  the  form  of  rectangular  solids  whose 
tops  are  on  the  same  level  are  connected  by  a  pipe  through  their 
bottoms.    The  base  of  one  is  6  inches  higher  than  that  of  the 
other.    Their  dimensions  are  4  x  5  x  2£  feet  and  4x7x3  feet 
respectively.    How  deep  is  the  water  in  the  larger  tank  when  the 
water  they  contain  equals  half  their  combined  capacity  ? 

191.  If  a  right  triangle  is  revolved  first  about  the  longer  leg 
and  then  about  the  shorter,  determine  in  which  case  (1)  the  vol- 
ume, (2)  the  lateral  area,  (3)  the  total  area,  is  the  greater. 

HINT.    Let  the  shorter  side  be  a ;  the  longer  side,  a  +  Ji.    Then 

F!  =  J  Tra2  (a  +  h)  =  $  TT (a3  +  a2/*). 

V2  =  i  TT  (a  +  A)2  a  =  £  TT  (a3  +  2  a"h  +  air}. 

192.  Find  (1)  the  volume  and  (2)  the  total  area  of  the  solid 
formed  by  the  rotation  of  an  equilateral  triangle  about  one  side. 

193.  The  tensile  strength  of  wire  is  proportional  to  the  area  of 
its  cross  section.    If  a  certain  wire  is  just  strong  enough  to  sup- 
port a  cube  of  iron  8  inches  on  a  side,  what  must  be  the  diameter 
of  the  wire  which  will  just  support  a  cube  of  iron  16  inches  on  a 
side  ? 

194.  An  irregular-shaped  body  is  placed  in  a  cylindrical  vessel 
of  water  whose  radius  is  r  inches.    The  water  rises  a  inches.   What 
is  the  volume  of  the  body  ? 

195.  The  volumes  of  two  cubes  are  in  the  ratio  of  4  :  5.    Find 
the  ratio  of  (1)  their  surfaces,  (2)  their  edges. 

196.  If  the  total  area  of  one  solid  is  twice  that  of  a  similar 
solid,  what  is  the  ratio  of  (1)  their  edges,  (2)  their  volumes  ? 

197.  The  dimensions  of  a  box  are  2x3x4  feet.    Find  the 
dimensions  of  a  box  of  the  same  shape  in  order  that  it  may  hold 
three  times  as  much. 

198.  The  bases  of  two  similar  pyramids  are  in  the  ratio  2  :  3. 
What  is  the  ratio  of  (1)  their  edges,  (2)  their  volume  ? 

199.  How  long  a  wire  T^  of  an  inch  in  diameter  can  be  drawn 
from  a  block  of  copper  2x4x6  inches  ? 


440  SOLID  GEOMETRY 

200.  Find  the  total  area  of  a  rectangular  solid  whose  base  is 
5x9  feet  and  whose  volume  is  900  cubic  feet. . 

201.  Eind  the  edge  of  a  cube  whose  total  surface  is  numerically 
equal  to  its  volume. 

202.  A  sector  having  an  angle  of  90°  is  cut  from  a  circle  8  inches 
in  radius,  and  the  edges  of  the  cut  are  joined  so  as  to  form  the 
lateral  area  of  a  cone.    Eind  the  volume  of  this  cone. 

203.  A  cone  of  revolution  has  a  slant  height  of  10  inches,  and 
its  radius  is  4  inches.    How  large  an  angle  must  be  cut  from  a 
circle  to  form  its  lateral  surface  ? 

204.  If  the  slant  height  of  a  cone  of  revolution  is  twice  the 
radius,  find  the  angle  that  must  be  cut  from  a  circle  in  order  to 
form  its  surface. 

205.  What  per  cent  of  a  square  stick  is  wasted  if  the  largest 
possible  cylindrical  stick  is  turned  out  of  it  on  a  lathe  ? 

'  206.  What  per  cent  of  the  volume  of  a  circular  cylindrical 
log  of  uniform  diameter  is  the  volume  of  the  greatest  square 
timber  which  can  be  cut  from  it  ? 

207.  The  slant  height  of  a  cone  of  revolution  makes  an  angle 
of  45°  with  the  base.    The  altitude  of  the  cone  is  25  inches.   Find 
the  volume. 

208.  A  pipe  f -inch  inside  measure  conducts  water  from  a  spring 
to  a  house  300  feet  distant.    It  is  desired  to  empty  the  pipe  after 
the  water  has  been  turned  off  at  the  spring.    Will  a  10-quart  pail 
hold  the  water  ? 

209.  A  tank  in  the  form  of  a  rectangular  solid  2x3x4  feet 
can  be  filled  through  a  pipe  f-inch  in  diameter  in  30  minutes. 
How  many  feet  of  water  flow  through  the  pipe  per  second  ? 

210.  It  is  desired  to  cut  off  a  piece  of  lead  pipe  2  inches  in 
outside  diameter  and  i-inch  thick,  so  that  it  will  melt  up  into  a 
cube  of  edge  6  inches.    How  long  a  piece  will  it  take  ? 

211.  The  volume  of  a  rectangular  solid  is  1296  cubic  inches, 
and  its  dimensions  are  in  the  ratio  1:2:3.    Find  the  dimensions. 


BOOK  VII  441 

212.  If  a  cone  and  a  cylinder  of  revolution  have  the  same  base 
and  equal  altitudes,  what  is  the  ratio  of  their  lateral  areas  ? 

213.  If  a  child  2^  feet  in  height  weighs  30  pounds,  what  would 
be  the  weight  of  a  man  6  feet  tall  of  the  same  proportions  ? 

214.  Which  is  the  more  heavily  built,  a  man  5^  feet  tall  who 
weighs  160  pounds,  or  one  6  feet  tall  who  weighs  200  pounds  ? 

215.  A  wooden  cone  of  revolution  has  a  diameter  of  6  inches 
and  an  altitude  of  10  inches.    An  auger  hole  1  inch  in  diameter 
is  bored  through  it,  the  axis  of  the  cone  and  the  auger  being  co- 
incident.   (1)  What  is  the  volume  of  the  small  cone  which  is  cut 
out  from  the  top  of  the  larger  cone  ?    (2)  What  is  the  volume  of 
the  cylindrical  hole  ? 

216.  The  radius  of  the  lower  base  of  a  frustum  of  a  cone  of 
revolution  is  twice  that  of  the  upper  base.    The  slant  height  is 
inclined  at  an  angle  of  45°  to  the  base.    If  the  altitude  of  the  frus- 
tum is  6  feet,  find  (1)  the  lateral  area,  (2)  the  volume,  of  the 
frustum. 

217.  A  circular  cone  is  4  feet  high.   The  shortest  and  the  longest 
elements  are  6  and  10  feet  respectively.    Find  the  volume. 

218.  How  many  board  feet  of  lumber  is  a  stick  4"  x  4"  at  one 
end,  2"  x  12"  at  the  other  end,  and  16'  long  ?    (One  board  foot 
is  1'  x  1'  X  1";  that  is,  it  contains  144  cubic  inches.) 

219.  If  a  lead  pipe   \  inch  thick  has  an  inner  diameter  of 
1^  inches,  find  the  number  of  cubic  inches  of  lead  in  a  piece  of 
pipe  12  feet  long. 

220.  How  many  cubic  yards  of  material  is   needed  for  the 
foundation  of  a  barn  40  x  80  feet  if  the  foundation  is  2  feet 
wide  and  12  feet  high  ? 

221.  A  pail  12"  high  is  in  the  form  of  a  frustum  of  a  cone.    If 
the  diameters  of  the  bases  are  10"  and  12"  respectively,  find  its 
capacity  in  quarts. 

222.  An  80-foot  flagpole  has  upper  and  lower  diameters  of  4 
and  16  inches  respectively.    Find  the  cost  of  painting  it  at  5  cents 
per  square  foot. 


442  SOLID  GEOMETRY 

223.  A  reservoir  is  in  the  form  of  an  inverted  square  pyramid 
with  bases  100  and  90  feet  on  a  side  respectively.    How  long 
will  it  require  an  inlet  pipe  to  fill  it  if  it  pours  in  400  gallons 
per  minute  ? 

224.  Compute  the  cost  of  the  lumber  necessary  to  resurface 
a  footbridge  16'  wide  and  150'  long  with  2"  plank  if  lumber  is 
$40  per  1000  board  feet. 

225.  A  druggist  sells  a  certain  kind  of  powder  in  a  rectangular 
box  4  x  2£  x  1£  inches  for  25  cents,  and  in  a  cylindrical  can 
2i  inches  high  and  2i  inches  in  diameter  for  20  cents.    Which  is 
it  more  economical  to  buy  ? 

226.  A  concrete  dam  is  4'  wide  at  the  top  and  12'  wide  at  the 
lowest  point.    It  is  72'  long ;  it  is  8'  high  at  one  end  and  12'  high 
at  the  other ;  at  the  lowest  point,  24'  from  the  latter  end,  it  is  21' 
high.   What  would  it  cost  to  build  it  at  |4.60  per  cubic  yard  ? 

227.  A  railway  embankment  across  a  valley  has  the  following 
dimensions  :  width  at  top,  20  feet ;  width  at  base,  45  feet ;  height, 
11  feet ;  length  at  top,  1020  yards ;  length  at  base,  960  yards. 
Find  its  volume. 


BOOK  VIII 


THE  SPHERE 
GENERAL  PROPERTIES  OF  THE  SPHERE 

589.  Sphere.    A  sphere  is  a  closed  surface  every  point  of 
which  is  equidistant  from  a  fixed  point  called 

the  center. 

590.  Radius.  A  line  connecting  any  point 
of  the  sphere  with  its  center  is  called  a  radius 
of  the  sphere. 

Since  the  sphere  is  closed,  every  point  in 
space  is  either  inside,  outside,  or  on,  the  sphere, 
according  as  its  distance  from  the  center  is  less  than,  greater  than, 
or  equal  to,  the  radius. 

Terms  such  as  diameter,  chord,  and  secant  are  used  in 
the  same  sense  for  the  sphere  as  they  are  for  the  circle. 

By  volume  of  the  sphere  we  mean  the 
volume  of  the  solid  inclosed  by  the  sphere. 

591.  Another  definition   of    the   sphere. 
A  sphere  may  also  be  defined  as  the  sur- 
face generated   by  the  complete   rotation 
of  a  semicircle  about  a  diameter. 

QUERY  1.    If    a   line   is   known  to  have   one 
point  inside  a  sphere,  in  how  many  points  must  C 

it  cut  the  sphere? 

QUERY  2.  The  distances  from  the  center  of  a  sphere  to  lines  which 
(1)  cut,  (2)  do  not  cut,  the  sphere  have  what  relation  to  the  length  of 
the  radius? 

443 


444  SOLID  GEOMETRY 

QUERY  3.  If  two  spheres  are  congruent,  is  it  certain  that  their  cen- 
ters coincide  when  the  spheres  coincide? 

QUERY  4.  If  two  spheres  are  congruent,  what  can  you  say  of  their 
radii  ? 

QUERY  5.  If  the  radii  of  two  spheres  are  equal,  are  the  spheres 
congruent  ? 

QUERY  6.  What  is  the  locus  of  points  five  inches  from  a  given  point  ? 

QUERY  7.  What  is  the  locus  of  points  two  inches  from  a  sphere 

which  is  six  inches  in  diameter? 

EXERCISES 

1.  Prove  that  any  diameter  of  a  sphere  is  twice  as  long  as  the 
radius. 

2.  Prove  that  a  diameter  is  longer  than  any  other  chord  of  a 
sphere. 

Theorem  1 

592.  If  a  plane  is  perpendicular  to  a  radius  of  a  sphere 
at  its  outer  extremity,  it  has  no  other  point  in  common 
with  the  sphere.  


Given  the  sphere  S  with  center  0,  and  the  plane  M  which  is 
perpendicular  to  the  radius  OP  at  P. 

To  prove  that          M  meets  S  only  at  P. 

Proof.         Let  K  be  any  point  in  M  other  than  P. 

Then  OK  >  OP.  §  422 

Therefore  K  is  outside  the  sphere.  §  590 

Therefore  no  point  of  M  except  P  is  on  S. 


BOOK  VIII  445 

593.  Tangency.    If  a  sphere  and  a  plane,  a  sphere  and  a 
line,  or  two  spheres  have  one  and  only  one  point  in  common, 
they  are  said  to  be  tangent  to  each  other. 

The  common  point  is  called  the  point  of  contact  in  each  case. 

594.  Corollary.    If  a  plane  is  tangent  to  a  sphere,  it  is  per- 
pendicular to  the  radius  drawn  to  the  point  of  contact. 

HINT.  Show  that  the  distance  from  the  center  of  the  sphere  to  the 
point  of  contact  is  shorter  than  any  other  line  from  the  center  to 
the  plane.  Then  apply  §  422. 

QUERY  1.    How  many  planes  are  tangent  to  a  sphere  at  a  given  point  ? 

QUERY  2.  What  is  the  locus  of  the  centers  of  spheres  of  fixed  radius 
which  are  tangent  to  a  given  plane  ? 

QUERY  3.  What  is  the  locus  of  the  centers  of  spheres  which  are  tan- 
gent to  both  faces  of  a  given  dihedral  angle  ? 

QUERY  4.  What  is  the  locus  of  the  centers  of  spheres  of  radius  two 
inches  which  are  tangent  to  both  faces  of  a  dihedral  angle  ? 

QUERY  5.  What  is  the  locus  of  the  centers  of  spheres  which  are 
tangent  to  a  plane  at  a  given  point? 

QUERY  6.  How  many  lines  are  there  in  each  tangent  plane  to  a  sphere 
which  have  one  and  only  one  point  in  common  with  the  sphere  ? 

QUERY  7.  On  what  kind  of  surface  do  all  of  the  tangent  lines  to  a 
sphere  from  a  fixed  exterior  point  lie  ?  If  the  point  approaches  very  close 
to  the  sphere,  what  does  the  surface  approach  in  form  ?  As  the  point 
recedes  farther  and  farther  from  the  sphere,  what  does  the  surface 
approach  in  form? 

EXERCISES 

3.  Prove  that  a  line  which  is  tangent  to  a  sphere  is  perpen- 
dicular to  the  radius  drawn  to  the  point  of  contact. 

4.  At  a  point  on  a  sphere  construct  a  plane  tangent  to  the 
sphere,  giving  a  reason  for  each  step. 

5.  If  two  spheres  are  tangent  to  a  plane  at  the  same  point, 
prove  that  their  line  of  centers  passes  through  the  point  of 
contact. 


446  SOLID  GEOMETRY 

Theorem  2 

595.  If  the  distance  from  the  center  of  a  sphere  to  a  plane 
is  less  than  the  radius  of  the  sphere,  the  plane  cuts  the 
sphere  in  a  circle  whose  center  is  the  foot  of  the  perpen- 
dicular drawn  from  the  center  of  the  sphere  to  the  plane. 


Given  any  sphere  S,  with  center  O  and  radius  r,  and  a  plane 
M  whose  distance  OC  from  O  is  less  than  r. 

To  prove  that  the  section  in  which  M  cuts  S  is  a  circle  with 
center  C. 

Proof.    Connect  any  two  points  of  the  section,  as  L  and  A',  with  C. 

Draw  OL  and  OK. 

A  OCL  is  congruent  to  A  OCR.  Why  ? 

Therefore  CK  =  CL.  Why  ? 

Hence  the  intersection  of  S  and  M  is  a  circle  of  which  C  is 
the  center.  §  154 

596.  Great  circle.    A  circle  on  a  sphere  whose  plane  passes 
through  the  center  of  the  sphere  is  called  a  great  circle. 

597.  Corollary  I.    All  great  circles  of  a  sphere  are  equal  and 
have  for  their  common  center  the  center  of  the  sphere. 

598.  Corollary  II.     Any  two  points  on  a  sphere   except  the 
extremities  of  a  diameter  determine  a  great  circle  of  the  sphere. 


BOOK  VIII  447 

599.  Corollary  III.    Any  three  points  on  a  sphere  determine  a 
circle  of  the  sphere. 

600.  Small  circle.    A  circle  on  a  sphere  whose  plane  does 
not  pass  through  the  center  of  the  sphere  is  called  a  small 
circle. 

QUERY  1.  What  can  you  say  of  the  distance,  from  the  center  of  a 
sphere,  of  a  plane  (1)  which  does  not  cut  the  sphere,  (2)  which  cuts  it, 
(3)  which  cuts  it  in  a  great  circle  ? 

QUERY  2.  If  a  plane  through  a  given  tangent  line  to  a  sphere  swings 
around  this  line  as  an  axis,  what  kind  of  sections  with  the  sphere  are 
obtained  ? 

QUERY  3.  What  relation  do  the  circles  obtained  in  Query  2  bear  to 
the  given  tangent  line  ? 

QUERY  4.  Is  a  given  tangent  line  to  a  sphere  tangent  to  all  the  great 
circles  and  also  to  all  the  small  circles  of  the  sphere  through  the  point 
of  contact  ? 

QUERY  5.  How  many  equal  circles  can  a  set  of  parallel  planes  cut 
from  a  sphere? 

QUERY  6.  On  MThat  kind  of  surface  are  all  the  radii  of  a  sphere 
which  meet  a  given  small  circle  ? 

QUERY  7.  What  is  the  intersection  of  a  circular  conical  surface  and 
a  sphere  whose  center  is  the  vertex  of  the  conical  surface  ? 

EXERCISES 

5.  A  great  circle  divides   the  surface   of  a  sphere  into  two 
congruent  parts. 

HINT.    Prove  by  superposition. 

6.  Any  two  great  circles  on  the  same  sphere  bisect  each  other. 

7.  Prove  that  the  distance  from  the  center  of  a  sphere  to  any 
one  of  its  tangent  lines  is  the  radius  of  the  sphere. 

HINT.   Pass  a  plane  and  apply  §  192. 

8.  Prove  that  the  distances  from  a  fixed  point  outside  a  sphere 
to  the  points  of  contact  of  the  lines  tangent  to  the  sphere  drawn 
through  that  point  are  all  equal. 


448 


SOLID  GEOMETRY 


9.  Prove  the  relation       1st  =  s2  -f-  d2, 

where  ?*,  s,  and  e£  denote,  respectively,  the  radius  of  a  sphere,  the 
radius  of  a  small  circle,  and  the  distance  of  the  plane  of  the  circle 
from  the  center  of  the  sphere.  From  this  relation  deduce  the 
following  properties  of  the  sphere  : 

(1)  The  radius  of  a  small  circle 
is  less  than  that  of  the  sphere. 

(2)  Circles  on  a  sphere  whose 
planes   are   equidistant  from  the 
center  of  the  sphere  are  equal. 

(3)  The  converse  of  (2). 

(4)  The  plane  of  the  larger  of  two  unequal  circles  of  a  sphere 
is  nearer  the  center  of  the  sphere  than  the  plane  of  the  smaller. 

(5)  The  converse  of  (4). 

601.  Hemisphere.     One  of  the   equal  parts  into  which  a 
sphere  is  divided  by  a  great  circle  is  called  a  hemisphere. 


Theorem  3 

602.  If  two  spheres  cut  each  other,  their  intersection  is 
a  circle  whose, plane,  is  perpendicular  to  their  line  of  centers. 


Given  two  spheres  S  and  S'  which  cut  each  other  and  whose 
centers  are  0  and  T  respectively. 

To  prove  that  their  intersection  is  a  circle  whose  plane  is  per- 
pendicular to  OT. 


BOOK  VIII  449 

Proof.  Through  any  point  P  of  the  intersection  pass  a  plane 
M  _L  to  the  line  of  centers  OT  at  K.  §  417 

Then  M  cuts  S  in  a  circle  whose  center  is  K  and  whose  radius 
is  KP.  §  595 

M  also  cuts  S1  in  a  circle  whose  center  is  K  and  whose  radius 
is  KP. 

These  two  circles  coincide,  since  they  lie  in  the  same  plane  and 
have  the  same  center  and  radius. 

Therefore  the  intersection  of  S  and  S'  is  a  circle. 

Since  M  is  _L  to  0  T  by  construction,  the  plane  of  this  circle  is 
_L  to  the  line  of  centers  of  S  and  S1. 

EXERCISES 

10.  The  radius  of  a  sphere  is  14  inches.    What  is  the  radius 
of  a  circle  whose  plane  is  4  inches  from  the  center  of  the  sphere  ? 

11.  Two  spheres  whose  centers  are  24  feet  apart  have  radii 
9   and   18  feet   respectively.    Find   the   area   of   the   circle   of 
intersection. 

12.  The  diameter  of  a  sphere  is  16  inches.  'Find  the  distance 
from  the  center  of  this  sphere  to  the  plane  of  a  circle  whose  area 
is  half  that  of  a  great  circle. 

603.  Inscribed  sphere.  If  a  sphere  is  tangent  to  each  of  the 
faces  of  a  polyhedron,  it  is  said  to  be  inscribed  in  the  polyhedron. 


604.  Circumscribed  sphere.  If  a  sphere  passes  through  each 
of  the  vertices  of  a  polyhedron,  it  is  said  to  be  circumscribed 
about  the  polyhedron. 


450 


SOLID  GEOMETRY 
Construction  1 


605.   Construct  a  sphere  circumscribed  about  a  given 
tetrahedron. 


Given  any  tetrahedron  PRST. 

To  construct  a  sphere  which  contains  the  points  P,  R,  S,  and  T. 

Construction.    At  O,  the  center  of  the  circle  circumscribed  about 
the  face  RST,  construct  a  line  OK  ±  to  RST.  §  420 

At  A,  the  middle  point  of  PR,  construct  a  plane  M  _L  to  PR. 

§416 

Then  the  point  of  intersection  C  of  OK  and  M  is  the  center  of 
a  sphere  which  passes  through  the  points  P,  R,  S,  and  T. 

Proof.  It  is  first  necessary  to  prove  that  the  line  OK  and  the 
plane  M  intersect. 

If  OK  were  II  to  M,  M  would  be  _L  to  the  face  RST.  §  438 

Therefore         RP  would  lie  in  the  plane  RST  §  440 
and            the  figure  PRST  would  not  be  a  tetrahedron. 
Hence        OK  and  M  must  meet  in  some  point  C. 

Since  C  lies  in  OK,  it  is  equidistant  from  R,  S,  and  T.  §  425 

Since  C  lies  in  M,  it  is  equidistant  from  P  and  R.  §  423 

Therefore  C  is  equidistant  from  the  four  points  P,  R,  S,  and  T, 
and  is  the  center  of  a  sphere  on  which  they  lie. 


BOOK  VIII  451 

QUERY  1.  How  many  points  are  required  to  determine  a  sphere  on 
which  they  all  lie  ? 

QUERY  2.  If  the  center  is  known,  how  many  additional  points  are 
necessary  to  determine  a  sphere  ? 

QUERY  3.   How  many  spheres  contain  a  given  circle? 

QUERY  4.  What  is  the  locus  of  the  centers  of  the  spheres  which 
contain  a  given  circle  ? 

QUERY  5.  How  many  spheres  can  be  passed  through  three  given 
points  ? 

QUERY  6.  What  is  the  locus  of  the  centers  of  the  spheres  which 
pass  through  two  given  points  ? 

QUERY  7.  What  is  the  locus  of  the  centers  of  the  spheres  with  given 
radius  which  pass  through  two  points  ? 

QUERY  8.  What  is  the  locus  of  points  at  a  given  distance  from  a 
given  point  and  equidistant  from  two  given  points  ? 

QUERY  9.  Two  points,  A  and  B,  are  a  distance  c  apart.  What  is 
the  locus  of  points  a  distance  m  from  A  and  n  from  Bl  Discuss  for 
all  cases. 

QUERY  10.  Under  what  circumstances  will  two  circles  in  different 
planes  determine  a  sphere  ? 

QUERY  11.  What  is  the  locus  of  points  a  given  distance  from  a 
given  point  and  at  the  same  time  equidistant  from  all  the  points  of 
a  given  circle  ?  Discuss  special  cases. 

QUERY  12.  What  is  the  locus  of  the  centers  of  the  spheres  which 
are  tangent  to  the  faces  of  a  given  dihedral  angle  ? 

EXERCISES 

13.  Construct  a  cube  circumscribed  about  a  given  sphere. 

14.  Construct  a  sphere  circumscribed  about  a  given  cube. 

15.  Prove  that  the  lines  which  are  perpendicular  to  the  faces 
of  a  tetrahedron  at  the  centers  of  their  circumscribed  circles  meet 
in  a  point. 

16.  Construct  a  sphere  of  given  radius  passing  through  three 
given  points. 

17.  Construct  a  sphere  inscribed  in  a  given  tetrahedron. 

18.  The  edge  of  a  regular  tetrahedron  is  8  inches.  What  is  the 
radius  of  the  circumscribing  sphere  ? 


452  SOLID  GEOMETRY 

MEASUREMENT  OF  THE  SPHERE 

Theorem  4 

606.  The  area  generated  by  the  base  of  an  isosceles  tri- 
angle rotated  about  an  axis  which  lies  in  its  plane  and 
contains  the  vertex,  but  which  does  not  cut  the  triangle, 
equals  the  product  of  the  projection  of  the  base  of  the  tri- 
angle on  the  axis  and  the  circumference  of  a  circle  whose 
radius  is  the  altitude  of  the  triangle. 

Case  I.  The  base  of  the  triangle  does  not  meet  and  is  not 
parallel  to  the  axis. 


Given  any  isosceles  triangle,  OAB,  whose  base  AB  does  not 
meet  and  is  not  parallel  to  the  line  RSy  which  lies  in  the  plane 
of  OAB  and  contains  the  vertex  of  OAB.  Let  MO  be  the  altitude 
of  the  triangle,  and  let  XY  be  the  projection  of  AB  on  RS.  And 
let  MZ  be  the  projecting  line  of  M,  the  midpoint  of  AB. 

To  prove  that  area  generated  by  AB  e'quals  XY  •  2  TrMO. 
Proof.  AX  is  _L  to  RS,  and  BY  is  _L  to  RS.  §  445 

Therefore  XABY  is  a  trapezoid  Why  ? 

and  generates  a  frustum  of  a  cone  when  rotated  about  R  S  as  an  axis. 
Therefore      area  generated  by  AB  =  AB  -  2  TrMZ.     (1)     §  559 
Draw  AC  II  to  RS. 


BOOK  VIII 


453 


In  AA/0Z  and  BAG,  MZ  is  _L  to  RS. 
MO  is  _L  to  AB. 
OZ  is  _L  to  BC. 

Hence  the  A  of  AMOZ  =  respectively  the  A  of  ABA  C 
and  &MOZ  and  BAC  are  similar. 

Hence  AB/MO  =  AC/MZ. 

That  is,  A B  -  MZ  =  A  C  -  MO  =  XY 

or  AB  •  2  TrMZ  =  XY>2  TrMO. 

Hence  area  generated  by  AB  =  XY  •  2  TrMO. 


MO, 


§445 

Why? 

Why? 

§  76 

Why? 

Why? 

Why? 

§124 

§32 


Case  II.    The  base  of  the  triangle  terminates  in  the  axis. 
HINTS.    Compare  the  triangles  MOZ  and  ABY,  as  in  Case  I,  apply 
§  556,  and  prove  that 

area  generated  by  AB  =  A  Y 

R 
X 


M 


Y 

S 

Case  III.    The  base  of  the  triangle  is  parallel  to  the  axis. 
HINTS.    Apply  §§  502,  500  and  prove  that 

area  generated  by  AB  =  XY  -  2  TrMO. 


454  SOLID  GEOMETRY 

Theorem  5 

607.  If  an  arc  of  a  semicircle  is  divided  into  a  number 
of  equal  parts,  and  chords  are  drawn  joining  the  points  of 
division  in  order %  and  the  figure  is  rotated  about  the  diam- 
eter of  the  semicircle  as  an  axis,  the  area  generated  by  the 
chords  equals  the  product  of  the  sum  of  their  projections 
on  the  axis  and  the  circumference  of  a  circle  whose  radius 
is  their  common  distance  from  the  center. 


Given  the  arc  AF.  Let  the  points  B,  C,  and  D  divide  it  into 
equal  parts ;  let  VW,  WX,  XY,  and  YZ  be  the  projections  of 
the  equal  chords  AB,  BC,  CD,  and  DF,  respectively,  on  the 
diameter  RS ;  and  let  MO  be  the  common  distance  of  the  chords 
from  the  center  0  of  the  circle. 

To  prove  that  the  area  generated  by  the  broken  line  ABCDF  is 
equal  to  VZ  -  2  TT M 0. 

Proof.    ABO,  ECO,  CDO,  and  DFO  are  equal  isosceles  A.    Why  ? 
The  projections  of  their  bases  on  RS  are  VW,  WX,  etc.     Given 

Area  generated  by  AB  equals  VW  -  2  irMO.  §  606 

Area  generated  by  BC  equals  WX  •  2  TrMO. 

Area  generated  by  CD  equals  XY  -2  TrMO. 

Area  generated  by  DF  equals  YZ  -  2  TrMO. 
Adding,  area  generated  by  broken  line  ABCDF  is  equal  to 
(VW+  WX+XY+  YZ}  •  2  irMO  =  VZ  •  2  TrMO. 


BOOK  VIII  455 

608.  Zone.    The  portion  of  a  sphere  included  between  two 
parallel  planes  which  intersect  the  sphere  is  called  a  zone. 

The  distance  between  the  planes  is 
called  the  altitude  of  the  zone. 

The  two  circular  sections  of  the  sphere 
which  are  made  by  the  parallel  planes 
are  called  the  bases  of  the  zone. 

609.  Zone  of  one  base.    If  one  of  the  parallel  planes  is  tan- 
gent to  the  sphere,  the  zone  is  said  to  have  one  base. 

If  both  planes  are  tangent  to  the  sphere,  the  zone  is  the  sphere 
itself,  and  the  altitude  of  the  zone  is  the  diameter  of  the  sphere. 

61.0.  Generation  of  a  zone.  Since  a  sphere  is  generated  by 
the  rotation  of  a  circle  about  a  diameter,  a  zone  is  generated 
by  the  rotation  of  an  arc  of  a  circle  about  a 
diameter  of  the  circle. 

If  the  arc  meets  the  axis  of  rotation,  the 
zone  has  only  one  base. 

If  the  arc  is  a  semicircle  rotating  about  its  diameter  the 
entire  sphere  is  generated. 

611.  Area  of  zone.  If  in  §607  the  number  of  chords  in- 
scribed in  the  arc  is  increased  without  limit  while  each 
becomes  very  short,  the  figure  generated  approaches  a  zone 
the  expression  for  the  area  of  which  may  be  inferred  from  the 
result  of  Theorem  5.  It  is  observed  that  as  the  number  of 
equal  chords  and  arcs  increases,  the  line  MO  of  Theorem  5 
increases  in  length  until  for  very  short  chords  it  becomes 
almost  equal  to  the  radius  of  the  sphere.  By  the  theory  of 
limits  it  is  possible  to  prove  the  following  statement;  but 
since  a  rigorous  proof  is  too  difficult  to  include  here,  and 
since  the  truth  of  the  proposition  is  evident  after  §  607,  we 
shall  assume  it  without  further  demonstration. 


456  SOLID  GEOMETRY 

Theorem  6 

612.  The  area  of  a  zone  equals  the  product  of  its  alti- 
tude and  the  perimeter  of  a  great  circle  of  the  sphere. 

613.  Corollary  I.    If  r  denotes  the  radius  of  a  sphere  and  h 
the  altitude  of  a  zone  whose  area  is  Z,  then 


614.   Corollary  II.    If  r  denotes  the  radius  of  a  sphere  and  S 
its  area,  then 


HI-NT.    In  §  613,  let  h  =  2  r. 

NOTE.  The  simplicity  of  this  expression  for  the  area  of  the  sphere 
is  one  of  the  most  remarkable  results  to  be  found  in  the  whole  field  of 
elementary  mathematics.  That  the  area  of  the  curved  surface  of  the 
sphere  should  turn  out  to  be  just  4  (not  4  plus  some  bothersome  irra- 
tional number)  times  the  area  of  the  largest  section  of  the  sphere  sug- 
gests a  harmonious  relationship  which  would  seem  even  more  astonishing 
if  the  symmetry  of  the  subject  had  not  led  us  to  take  such  results  as  a 
matter  of  course. 

QUERY  1.  If  different  diameters  are  used  as  axes,  will  a  given  arc 
always  generate  zones  with  the  same  altitude  ? 

QUERY  2.  In  what  position  do  you  think  that  the  axis  should  lie  in 
order  that  a  given  arc  may  generate  (1)  the  zone  of  least  area,  (2)  the 
zone  of  greatest  area  ? 

EXERCISES 

In  the  following  exercises  r  and  S  denote  the  radius  and  sur- 
face, respectively,  of  a  sphere,  and  Z  denotes  the  area  and  h  the 
altitude  of  a  zone. 

19.  Given  r  =  4.    Find  S. 

20.  Given  r  =  1/2.    Find  S. 

21.  Given  S  ==  STT.    Find  r. 

22.  Given  r  =  6,  h  =  2.    Find  Z. 

23.  Given  r  =  9.4,  k  =  5J.    Find  Z. 


BOOK  VIII  457 

24.  Given  S  =  72  TT,  h  =  2.    Find  Z. 

25.  Given  Z  =  irr.    Find  h. 

26.  On  the  same  sphere  or  equal  spheres  the  areas  of  two  zones 
are  in  the  same  ratio  as  their  altitudes. 

27.  The  areas  of  two  spheres  are  in  the  same  ratio  as  the  squares 
of  their  radii. 

28.  The  area  of  a  zone  of  one  base  equals  the  area  of  the 
circle  whose  radius  is  the  chord  of  the  generating  arc  of  the  zone. 

HINT.  To  prove  that  Z  =  irPA  2,  pass  a  section 
through  PA  making  a  great  circle,  and  consider 
the  triangle  PP'A. 

29.  A  zone  of  one  base  is  36  TT  square  inches 
in  area,  and  the  chord  of  its  generating  arc  is 
4  inches  from  the  center  of  the  sphere.    Find 
the  surface  of  the  sphere. 

30.  Find  the  area  of  the  greatest  zone  which 

can  be  generated  by  an  arc  whose  chord  is  9  inches  long,  on  a 
sphere  of  radius  16  inches. 

31.  Prove  that  half  the  surface  of  the  earth  is  included  between 
the  parallels  30°  K  and  30°  S. 

32.  What  portion  of  a  sphere  is  visible  from  an  exterior  point 
at  a  distance  from  it  equal  to  the  radius  ? 

33.  The  radius  of  a  sphere  is  5.    Find  the  radius  of  a  sphere 
having  an  area  three  times  that  of  the  given  sphere. 

34.  The  radius  of  a  given  sphere  is  8.    Find  the  radius  of  a 
sphere  having  an  area  one  half  that  of  the  given  sphere. 

35.  The  radius  of  a  given  sphere  is  10  inches.    What  is  the 
altitude  of  a  zone  whose  area  is  one  fourth  that  of  the  sphere  ? 

36.  The  radius  of  a  given  sphere  is  r.    What  is  the  radius  of 
a  zone  of  one  base  whose  area  is  one  third  that  of  the  sphere  ? 

37.  The  radius  of  a  given  sphere  is  r.    A  zone  whose  area  is 
one  fourth  that  of  the  sphere  has  one  base  twice  the  radius  of  the 
other.    How  far  from  the  center  of  the  sphere  is  the  larger  base  ? 


458 


SOLID  GEOMETRY 
Theorem  7 


615.   The  volume  of  a  sphere  equals 
the  radius  of  the  sphere. 


4-rrr* 


where  r  denotes 


FIG.  1 


FIG.  2 


Given  any  sphere  of  radius  r  and  volume  V. 
To  prove  that  V= 


Proof.  Consider  the  figure  consisting  of  a  cylinder  of  revolu- 
tion the  radius  of  whose  base  is  r  and  whose  altitude  is  also  r, 
having  had  removed  from  it  the  cone  of  revolution  whose  base  is 
the  upper  base  of  the  cylinder  and  whose  vertex  is  at  the  center 
of  the  lower  base. 

Let  the  hemisphere  of  radius  r  have  its  base  in  the  same  plane 
with  the  base  of  the  other  figure.  Pass  a  plane  through  both 
figures  parallel  to  the  common  plane  of  their  bases  at  any  dis- 
tance PR  =  OK  from  the  base.  This  plane  will  cut  the  sphere 
in  a  circle  whose  radius  is  RT,  and  the  other  figure  in  a  ring 
whose  outer  radius  is  KS  and  whose  inner  radius  is  KC. 


In  Fig.  1,  RT'2  =  PT2  -  PR' 

In  Fig.  2,  OA/AB  =  OK/KC. 
But  OA  =  AB. 

Hence,  OK  =  KC  =  PR. 

Also  PT=KS  =  r. 


(1)    §284 

§263 

Given 

Given 


BOOK  VIII  459 

Substituting  KS  for  PT  and  KC  for  PR  in  (1), 


or  7TR  T2  =  TrKS2  -  7TKC*. 

The  left-hand  member  of  this  equation  is  the  area  of  the  circle 
cut  from  the  sphere,  while  the  right-hand  member  is  the  area  of 
the  ring  in  Fig.  2. 

Hence  the  area  of  a  plane  section  of  Fig.  1  at  any  distance  from 
the  base  equals  the  area  of  a  plane  section  of  Fig.  2  at  the  same 
distance  from  its  base. 

Therefore  the  volume  of  the  hemisphere  equals  the  volume  of 
the  cylinder  with  the  cone  removed.  §  481 

But  the  volume  of  the  cylinder  minus  that  of  the  cone 
=  Trr2  •  r  -  1/3  irr2  .  r  =  2/3  Trr3. 

Therefore  the  volume  of  the  hemisphere  =  2/3  Trr3, 

and  the  volume  of  the  entire  sphere  equals  4/3  Trr3.   §  §  498,  550 

EXERCISES 

38.  The  radius  of  a  sphere  is  4  inches.    Find  (1)  the  surface, 
(2)  the  volume. 

39.  The  surface  of  a  sphere  is  616  square  inches.    Find  the 
volume. 

40.  A  zone  whose  altitude  is  6  inches  has  an  area  of  3696 
square  inches.    Find  (1)  the  area,  (2)  the  volume  of  the  sphere. 

41.  A  zone  of  one  base  is  36  TT  square  inches  in  area,  and  the 
chord  of  its  generating  arc  is  4  inches  from  the  center  of  the 
sphere.    Find  the  volume  of  the  sphere. 

42.  Prove  that  the  volumes  of  any  two  spheres  are  propor- 
tional to  the  cubes  of  their  radii. 

43.  If  one  sphere  has  twice  the  surface  of  another,  find  the 
ratio  of  their  volumes. 

44.  If  one  sphere  has  twice  the  volume  of  another,  find  the 
ratio  of  their  surfaces. 


460 


SOLID  GEOMETRY 


616.  Spherical  segment.    The  solid  bounded  by  a  zone  and 
the  planes  of  its  bases  is  called  a  spherical  segment  of  two  bases. 


The  altitude  of  a  spherical  segment  is  the  altitude  of  its 
zone. 

If  the  zone  has  only  one  base,  the  segment  is  said  to  have 
one  base. 

617.  Spherical  sector.    The  solid  generated  by  the  rotation 
of  a  sector  of  a  circle  about  an  axis  which  passes  through 
the    center    of    the   circle, 

but  which  does  not  cut  the 
sector,  is  called  a  spherical 
sector. 

The    bounding    surfaces 
of  a  spherical  sector  are  a 
zone,  which  is   called  the 
base  of  the  sector,  and  either  one  or  two  conical  surfaces, 
according  as  the  zone  has  one  or  two  bases. 

618.  Spherical  cone.    If  the  base  of  a  spherical  sector  is  a 
zone  of  one  base,  the  sector  consists  of  a  cone  and  a  spherical 
segment  of  one  base.    This  figure  is  some- 
times called  a  spherical  cone. 

The  spherical  sector  in  the  adjacent  figure 
may  be  looked  upon  either  as  a  spherical  seg- 
ment with  two  cones  removed  or  as  the  entire 
solid  sphere  with  two  spherical  cones  removed. 


BOOK  VIII  461 

619.  Volume  of  spherical  cone.   Consider  the  pyramids  whose 
common  vertex  is  the  center  of  the  sphere  and  whose  bases 
have  their  vertices  on  the  base  of  the  spherical  cone.    These 
pyramids  can  be  taken  numerous  enough,  and  their  bases  can 
each  be  taken  small  enough,  so  that  each  altitude  is  nearly 
equal  to  the  radius  of  the  sphere,  and  the 

sum  of  their  bases  is  almost  exactly  equal 
to  the  base  of  the  spherical  cone.  The  sum 
of  the  pyramids  themselves  forms  a  solid 
approximating  the  spherical  cone  as  closely 
as  may  be  desired.  Now  the  volume  of  each 
pyramid  equals  one  third  the  product  of  its 
base  and  altitude,  and  it  can  be  proved  that  as  the  number 
of  pyramids  becomes  very  large  and  each  of  their  bases 
becomes  very  small,  the  volume  of  the  sum  of  the  pyramids 
approaches  one  third  the  product  of  the  base  of  the  spherical 
cone  and  the  radius.  But  since  the  sum  of  the  pyramids 
approaches  the  spherical  cone,  we  may  infer  the  truth  of  the 
following  theorem,  a  rigorous  proof  of  which  is  beyond  the 
scope  of  this  book. 

Theorem  8 

620.  The  volume  of  a  spherical  cone  equals  one  third 
the  product  of  the  area  of  the  zone  ivhich  forms  its  base, 
and  the  radius  of  the  sphere. 

621.  Corollary  I.    The  volume  of  any  spherical  sector  equals 
one  third  the  product  of  the  area  of  the  zone  ivhich  forms  its 
base,  and  the  radius  of  the  sphere. 

622.  Corollary  II.    If  h  denotes  the  altitude  of  the  zone  which 
forms  the  base  of  a  spherical  sector  on  a  sphere  of  radius  r,  the 

volume,  F,  of  the  sector  is  V=  — 

o 


462  SOLID  GEOMETRY 

623.  Corollary  III.    The  volume  of  a  spherical  segment  of  one 
base  is 


where  h  is  the  altitude  of  the  segment  and  r  is  the  radius  of  the 
sphere. 

HINTS.    The  segment  is  equal  to  a  spherical  cone  less  the  ordinary 
cone  whose  base  is  the  base  of  the  segment  and 
whose  vertex  is  at  the  center  of  the  sphere.    Let  a 
denote  the  radius  of  the  base  of  the  segment. 


Thei  2,rr2V    7m2(r-/Q 

3  3 

But  a2  =  h(2r-h).  §282 

Hence     V=  J[2  r*k  -  h  (2  r  -  h)  (r  -  /*)]  =  —(3  r  -  A). 
o  3 

QUERY  1.  Explain  how  the  solid  bounded  by  a  hemisphere  and  a 
plane  can  be  considered  as  a  special  case  either  of  a  spherical  segment 
or  of  a  spherical  sector. 

QUERY  2.  What  is  the  locus  of  the  centers  of  spheres  of  given  radius 
which  are  tangent  to  two  given  intersecting  planes? 

QUERY  3.  What  relation  must  exist  between  the  lengths  of  the  radii 
and  the  lengths  of  the  line  of  centers  of  two  spheres  in  order  that  one 
may  lie  entirely  inside  the  other  ? 

QUERY  4.    What  are  some  examples  of  great  circles  on  the  earth  ? 

QUERY  5.    What  are  some  examples  of  small  circles  on  the  earth  ? 

QUERY  6.  If  a  sphere  is  viewed  from  a  finite  distance,  can  the  observer 
see  an  entire  great  circle  ? 

QUERY  7.  What  kind  of  figure  is  the  shadow  of  a  sphere  cast  on  a 
horizontal  plane  by  the  sun  when  it  is  directly  overhead? 

EXERCISES 

45.  Find  the  volume  of  the  segment  on  a  sphere  which  is  cut 
off  by  a  plane  2  inches  from  the  center  of  the  sphere  whose  radius 
is  6  inches. 

46.  What  is  the  volume  of  a  segment  of  one  base  of  a  sphere 
whose  radius  is  r  if  its  altitude  is  one  half  that  of  r  ? 


BOOK  VIII  463 

47.  The  water  in  a  hemispherical  bowl  18  inches  across  the  top 
is  6  inches  deep.    What  per  cent  of  the  capacity  of  the  bowl  is 
filled  ? 

48.  A  4-inch  auger  hole  is  bored  through  a  10-inch  sphere,  the 
axis  of  the  hole  coinciding  with  a  diameter  of  the  sphere.    Find 
the  volume  remaining. 

REVIEW  EXERCISES 

49.  Prove  that  two  circles  which  have  two  points  in  common, 
but  which  do  not  lie  in  the  same  plane,  determine  a  sphere. 

50.  Prove  that  a  circle  and  a  point  not  in  its  plane  determine  a 
sphere. 

51.  Prove  that  the  surface  of  a  sphere  and  the  lateral  area  of 
the  circumscribed  cylinder  of  revolution  are  equal. 

52.  Find  the  ratio  of  the  volume  of  a  sphere  to  that  of  the  cir- 
cumscribed cylinder  of  revolution. 

53.  A  piece  of  lead  pipe  is  50  feet  long.    Its  outer  radius  is 
2  inches,  and  it  is  ^-inch  thick.    Into  how  many  spherical  bullets 
^-inch  in  diameter  can  it  be  melted  ? 

54.  A  cylinder  of  revolution  is  capped  on  each  end  by  a  hemi- 
sphere.   Show  that  the  total  surface  of  the  figure  equals  the  prod- 
uct of  its  entire  length  and  the  circumference  of  the  base  of  the 
cylinder. 

55.  Find  the  ratio  of  the  volumes  of  a  sphere  and  a  cube  if 
their  surfaces  are  equal. 

56.  Through  a  point  6  inches  from  a  sphere  of  radius  4  inches 
all  the  tangent  lines  to  the  sphere  are  drawn.    Find  the  lateral 
area  of  the  conical  surface  included  between  the  point  and  the 
sphere. 

57.  The  volume  and  the  surface  of  a  sphere  are  expressed  by 
the  same  number.    Find  the  radius. 

58.  Find  the  volume  of  a  cone  whose  vertex  angle  is  60°  and 
which  is  inscribed  in  a  sphere  whose  radius  is  10  inches. 


464  SOLID  GEOMETRY 

59.  The  outside  diameter  of  a  spherical  iron  shell  2  inches 
thick  is  14  inches.    Find  its  weight  if  a  cubic  inch  of  iron  weighs 
4.2  ounces. 

60.  A  wooden  sphere  whose  radius  is  15  inches  rests  in  a  cir- 
cular hole  in  a  board  the  radius  of  which  is  5  inches.    How  far 
below  the  upper  surface  of  the  board  does  the  sphere  extend  ? 

61.  Find  the  volume  of  a  cube  inscribed  in  a  sphere  of  radius  r. 

62.  Find  the  volume  of  a  regular  octahedron  inscribed  in  a 
sphere  of  radius  r. 

63.  Prove  that  two  lines  tangent  to  a  sphere  at  the  same  point 
determine  the  tangent  plane  to  the  sphere  at  that  point. 

64.  Referring  to  the  Prismoidal  Formula  in  §  588,  prove  that 
the  volume  of  a  sphere  is  given  by  that  Formula. 

HINT.    The  areas  of  the  two  extreme  sections  are  each  zero,  while  the 
mid-section  is  a  great  circle. 

65.  Prove  that  the  volume  of  a  spherical  segment  of  one  base 
can  be  found  by  the  Prismoidal  Formula. 

66.  A  cylindrical  glass  of  radius  1.5  inches  and  altitude  6  inches 
is  filled  with  water  to  a  depth  of  2  inches.    If  three  spheres  each 
1  inch  in  diameter  are  dropped  into  the  glass,  by  how  much  is  the 
level  of  the  water  raised  ? 

GEOMETRY  ON  THE  SPHERE 

624.  Practical  importance  of  spherical  geometry.    We  shall 
now  study  briefly  the  properties  of  figures 

drawn  on  a  sphere.  The  fact  that  in  the 
sciences  of  Geodesy,  Astronomy,  Naviga- 
tion, and  to  a  certain  extent  Civil  Engineer- 
ing, the  theorems  of  Spherical  Geometry 
find  important  application  gives  practical 
significance  to  this  part  of  our  subject. 

625.  Spherical  polygon.    The  portion  of  a  sphere  bounded 
by  arcs  of  great  circles  is  called  a  spherical  polygon. 


BOOK  VIII  465 

626.  Spherical  angle.    The  figure  formed  on  a  sphere  at  the 
point  where  two  arcs  of  great  circles  meet  p\ 

each  other  is  called  a  spherical  angle.  ^^g^T 

627.  Measure  of  a  spherical  angle.    The 
numerical  measure  of  a  spherical  angle  is 
equal   to    the   numerical    measure    of    the    l 
angle  between   the  tangents  to  the  great 
circles  at  their  point  of  intersection. 

In  this  text  it  is  not  necessary  to  define  or  to  discuss  the  angle 
between  two  small  circles  or  other  curves  which  may  be  drawn 
on  the  sphere. 

Terms  which  are  used  in  spherical  geometry  in  the  same  sense 
as  in  plane  geometry  will  not  be  defined  again. 

QUERY.  Does  it  make  any  difference  which  point  of  intersection  of 
two  great  circles  is  taken  in  denning  the  measure  of  the  angle  between 
them? 

EXERCISES 

67.  Prove  that  the  angle  between  two  great  circles  is  equal  to 
the  angle  between  their  planes. 

68.  Construct  the  arc  of  a  great  circle  making  an  angle  of 
(a)  30°,  (b)  90°,  (c)  45°  with  a  given  great  circle  at  a  given  point. 

628.  Relation  between  spherical  polygons  and  polyhedral 
angles.    If  radii  of  the  sphere  are  drawn  from  the  vertices 
of    a   spherical   polygon,    and    the    planes 

determined  by  successive  pairs  of  these 
radii  are  passed,  a  polyhedral  angle  is 
formed  which  is  intimately  related  to  the 
polygon. 

Prove    each    of   the    following    proposi- 
tions relating  to  the  spherical  polygon: 

629.  The  face  angles  of  a  polyhedral  angle  are  measured  by  the: 
arcs  which  form  the  sides  of  the  corresponding  spherical  polygon. 


466 


SOLID  GEOMETRY 


630.  The  dihedral  angles  of  a  polyhedral  angle  are  equal  to 
the  angles  of  the  corresponding  spherical  polygon. 

HINT.   Apply  §§  627  and  432. 

631 .  The  sum  of  the  sides  of  a  convex  spherical  polygon  is 
less  than  a  great  circle. 

HINT.   Apply  §§  629,  572. 

632.  Any  side  of  a  spherical  triangle  is  less  than  the  sum  of 
the  other  two  sides. 

HINT.    Apply  §  570. 

633.  Congruence.    Two  spherical  triangles  are  congruent  if 
their  sides  and  their  angles  are  equal  each 

to  each  and  arranged  in  the  same  order. 

As  in  the  case  of  all  other  geometric  fig- 
ures, if  two  spherical  triangles  are  identical 
in  every  respect,  they  may  be  looked  upon 
as  merely  different  positions  of  the  same 
figure  and  may,  by  §  20,  be  superposed. 

634.  Symmetric  spherical  triangles.   Two  spherical  triangles 
are    symmetric   if   their   parts    are    equal    each   to   each   but 
arranged  in  opposite  orders  when  both 

triangles  are  viewed  from  the  center  of 
the  sphere. 

From  an  inspection  of  the  figure  it  C'j 
appears  that  the  triangles  determined  by 
vertical  trihedral  angles  (§  580)  whose 
vertex  is  at  the  center  of  the  sphere  are 
symmetric,  since  their  angles  and  their 
sides  are  equal  each  to  each  (§§  629,  630)  and  the  corre- 
sponding parts  are  arranged  in  opposite  orders  when  viewed 
from  the  center  of  the  sphere, 


BOOK  VIII  467 

Theorem  9 

635,  Two  spherical  triangles  on  the  same  sphere  or  on 
equal  spheres,  which  have  the  three  sides  of  one  equal  respec- 
tively to  the  three  sides  of  the  other,  are  either  congruent 
or  symmetric,  according  as  the  equal  sides  are  arranged 
in  the  same  or  in  opposite  orders. 


Case  I.    When  the  equal  sides  are  arranged  in  the  same  order. 

Given  the  two  spherical  triangles  ABC  and  A'B'C',  in  which 
the  corresponding  sides  are  equal  and  are  arranged  in  the 
same  order. 

To  prove  that     A  ABC  is  congruent  to  AA'B'C'. 

Proof.  Construct  the  trihedral  A  corresponding  to  ABC  and 
A'B'C'. 

Then  the  face  A  of  0-ADC  are  equal  to  the  face  A  of  0-A'B'C1 
and  are  arranged  in  the  same  order.  §  629 

Hence  the  dihedral  A  OA  =  OA',  OB  =  OB',  OC  =  OC',      §  569 

anaZ.ABC=^A'B'C',^ACB=Z.A'C'B',Z.BAC=Z.B'A'C1.  §630 

Therefore  AABC  is  congruent  to  AA'B'C'.  §  633 

Case  II.     When  the  equal  sides  are  arranged  in  opposite  order. 

Given  the  two  spherical  triangles  ABC  and  A'B'C',  whose  corre- 
sponding sides  are  equal  and  arranged  in  opposite  order. 

To  prove  that     AABC  is  symmetric  to  AA'B'C'. 

Proof.  The  demonstration  is  identical,  except  that  the  tri- 
hedral angles,  and  therefore  the  triangles,  are  symmetric.  §  634 


468  SOLID  GEOMETRY 

Theorem  10 

636.   The  angles  opposite  the  equal  sides  of  an  isosceles 
spherical  triangle  are  equal. 

A 


Given  the  spherical  triangle  ABC  having  the  side  AB  equal 
to^C. 

To  prove  that  Z_fi  =  Z.C. 

Proof.    Construct  the  mid-point  of  the  arc  BC  and  denote  it  by  M. 
Let  AM  IQQ  the  arc  of  the  great  circle  determined  by  A  and  M. 

§598 
Then  the  spherical  A  A  MB  and  AMC1  are  symmetric.         §  634 

,    Therefore  Z7J  =  ZC.  §635 


QUERY  1.  Are  isosceles  spherical  triangles  whose  parts  are  equal 
each  to  each  necessarily  congruent  ? 

QUERY  2.  Are  isosceles  spherical  triangles  whose  parts  are  equal 
each  to  each  necessarily  symmetric  ? 

QUERY  3.  If  you  try  to  superpose  two  symmetric  spherical  triangles 
by  turning  one  of  them  over  so  that  the  equal  parts  are  arranged  in  the 
Same  order,  why  are  you  unable  to  do  it?  Is  the  same  difficulty  met  in 
the  case  of  plane  triangles  ? 

QUERY  4.  If  two  triangles  are  symmetric  to  the  same  triangle,  are 
they  necessarily  congruent? 

QUERY  5.  If  from  any  point  of  AM  in  the  figure  above  two  arcs  of 
great  circles  are  drawn  to  B  and  C  respectively,  are  these  arcs  neces- 
sarily equal  ? 

QUERY  6.  Can  A  in  the  figure  above  be  so  situated  that  the  angles 
B  and  C  are  both  right  angles  ? 


BOOK  VIII 


469 


637.  Poles.    The  poles  of  a  circle  on  a  sphere  are  the  points 
where    a  tine   perpendicular    to   the   plane 

of  the  circle  at  its  center  intersects  the 
sphere. 

QUERY  1.  How  many  poles  does  a  circle  on  a 
sphere  have? 

QUERY  2.  How  can  one  obtain  a  set  of  circles 
on  a  sphere  which  have  the  same  poles  ? 

QUERY  3.  Can  two  great  circles  have  the  same 
poles  ? 

EXERCISES 

69.  Prove  that  every  circle  of  a  sphere  through  the  poles  of  a 
given  circle  is  a  great  circle. 

70.  Every  point  on  a  circle  of  a  sphere  is  equidistant  from  a 
pole  of  that  circle. 

Theorem  11 

638.  The  arcs  of  the  great  circles  joining  any  point  of  a 
small  circle  to  one  of  its  poles  are  equal. 


HINT.    Apply  §§25,  179. 

QUERY  4.    What  kind  of  a  spherical  triangle  is  PAB? 

639.  Corollary.  The  points  on  a  sphere  ivhich  are  a  constant 
distance  from  a  fixed  point  on  the  sphere  lie  on  a  circle  of  the 
sphere  of  which  the  fixed  point  is  the  pole. 

HINTS.  Let  P  be  the  fixed  point  and  PA  be  the  constant  distance. 
Pass  the  plane  through  A  which  is  perpendicular  to  the  radius  PO, 
and  apply  §§  425  (1),  595. 


470 


SOLID  GEOMETRY 


640.  Constructions  on  the  sphere.    It  follows  from  §  639  that 
if  the  point  of  a  pair  of  compasses  is  placed  on  a  point  of  a 
sphere  and  a  continuous  curve  is  drawn  on 

the  sphere  with  the  aid  of  the  compasses, 
this  curve  will  be  a  circle  of  the  sphere.  If 
it  is  intended  to  perform  the  constructions 
of  spherical  geometry  by  operations  that  can 
be  carried  out  on  the  surface  of  the  sphere, 
this  method  of  drawing  circles  must  replace 
their  determination  by  the  passing  of  planes. 
A  method  of  drawing  a  great  circle  deter- 
mined by  two  given  points  is  found  in  §  654. 

Construction  2 

641.  Construct  the  diameter  of  a  sphere  from  measure- 
ments on  its  surface. 


.  2 


FIG.  3 


Given  a  sphere  S. 

To  construct  its  diameter  from  measurements  on  its  surface. 

Analysis.  If  K  is  a  small  circle  on  S  and  its  poles  are  P  and  T, 
one  observes  that  the  triangle  PA  T  is  a  right  triangle,  and  that 
AR  is  the  altitude  on  the  hypotenuse  PT.  If  we  can  construct 
AP  and  AR,  the  triangle  APR  and  hence  APT  can  be  constructed. 
The  diameter  PT  will  then  be  found. 


BOOK  VIII 


471 


Construction.  Set  the  compasses  with  the  distance  AP  between  the 
points  and  construct  K,  a  small  circle  of  the  sphere  (Fig.  1).  §  639 

With  the  compasses  determine  the  lengths  of  the  chords  AB, 
BC,  and  CA,  which  join  any  three  points  of  K,  as  A,  B,  and  C. 

In  a  plane  construct  the  triangle  A^B^C^  whose  sides  are  equal 
respectively  to  AB,  BC,  and  CA  (Fig.  2).  §  232 

Circumscribe  the  circle  K  about  A1B1C1  and  denote  its  center 


Construct  a  plane  right  triangle  P^A^T^  (Fig.  3)  having  one  leg, 
v  equal  to  AP,  and  the  altitude  on  the  hypotenuse,  A^R^  equal 


to  Afl^ 

Proof. 

Therefore 

Also 

Now 

Therefore 

But 
and 

Therefore 
and 

Hence 


P2T2  is  the  required  diameter. 

PT  is  J_  to  plane  of  circle  K. 

AR  is  the  altitude  on  PT. 

PA  T  is  a  rt.  Z. 
AA1B1C1  is  congruent  to  A  ABC. 


2R2  is  congruent  to  A  PAR, 
A  P2A2T2  is  congruent  to  A  PA  T. 
T  is  equal  to  PT,  the  diameter  of  S. 


§  637 

§  414 

§  217 

§  33 

§27 

Why? 

§  97 

Why  ? 


642.  Polar  distance.    The  length  of  the  arc  of  a  great  cir- 
cle joining  any  point  of  a  great  or  a  small  circle  of  a  sphere 
to  the  nearer  pole  of  the  circle  is  called  p 

the  polar  distance  of  the  circle. 

Thus,  in  the  figure  the  arc  AP  is  the      ^ 
polar  distance  of  the  circle. 

643.  Corollary.    Two    equal   circles   on   a 
sphere  have  equal  polar  distances. 

HINT.  Prove  by  superposition. 


472 


SOLID  GEOMETRY 


Theorem  12 
644.  Two  symmetric  spherical  triangles  are  equal  in  area. 

A  A' 


Given  the  two  symmetric  spherical  triangles  ABC  and  A  'B'  C', 
that  is,  two  spherical  triangles  whose  parts  are  equal  each  to 
each  but  arranged  in  opposite  orders. 

To  prove  AABC  =  AA'B'Cf. 


Proof.  Pass  the  planes  determined  by  the  vertices  of  ABC  and 
A'B'C'j  respectively,  forming  small  circles  S  and  S',  in  which  the 
given  spherical  A  are  inscribed. 

Since  AB=A'B'>  AC=A'C',  BC  =  B'C',  Hyp. 

the  sides  of  the  plane  A  ABC  and  A'B'C'  are  equal.     §  178 
Hence  the  two  small  circles  are  equal.  Why  ? 

Let  P  and  P'  be  the  poles  of  S  and  S'  respectively. 

Then  the  spherical  APAB,  PAC,  PCS,  P'A'B',  P'A'C',  P'C'B', 
are  all  isosceles.  §  638 

Therefore 

APAC=AP'A'C',  APAB=AP'A'B',  PBC=AP'B'C'.  §635 
Adding, 

APAC+APAB+APBC=AP'A'C'+AP'A'B' 
or  AABC=AA'B'C'. 


BOOK  VIII  4T3 

645.  Relation  between  plane  and  spherical  geometry.    The 

subject  of  plane  geometry  consists  in  proving  propositions 
which  follow  from  the  assumptions  and  definitions  there  laid 
down.  We  have  also  a  spherical  geometry,  in  which  the  fig- 
ures are  drawn  not  on  a  plane  but  on  a  sphere.  In  order 
to  understand  the  similarities  and  the  contrasts  between  plane 
geometry  and  spherical  geometry,  it  is  necessary  to  determine 
what  figures  on  the  sphere  correspond  to  the  line  and  to  the 
circle  on  the  plane,  and  to  find  out  how  closely  the  assump- 
tions made  regarding  the  properties  and  relations  of  these 
fundamental  elements  in  the  plane  may  be  carried  over  to 
spherical  geometry. 

We  have  already  seen  (§  640)  that  a  small  circle  on  a 
sphere  can  be  drawn  with  compasses  and  therefore  corre- 
sponds to  a  circle  in  the  plane.  In  fact,  a  small  circle  is  the 
curve  on  the  sphere  such  that  the  distances  to  any  of  its 
points  from  the  pole  of  the  circle  are  equal. 

The  two  most  important  properties  of  the  line  in  plane 
geometry  are  the  following: 

1.  A  line  is  determined  by  any  two  of  its  points. 

2.  The  shortest  path  between  two  points  of  a  plane  is 
along  the  line  joining  them. 

From  §  598  it  follows  that  a  great  circle  is  determined  by 
any  two  of  its  points  unless  those  points  lie  at  the  extremi- 
ties of  a  diameter.  With  the  exception  noted  the  great  circle 
has  property  1  of  the  line  in  a  plane. 

The  same  exception  appears  when  we  make  the  statement 
regarding  great  circles  corresponding  to  the  fact  that  two 
lines  in  a  plane  never  have  two  points  in  common,  for  two 
great  circles  never  have  two  points  in  common  except  those 
points  which  are  the  extremities  of  a  diameter. 

We  shall  now  show  that  (2)  corresponds  to  a  property  of  the 
great  circle. 


474  SOLID  GEOMETRY 

Theorem  13 

646.  The  minor  arc  of  the  great  circle  joining  two  points 
on  a  sphere  is  the  shortest  curve  on  the  sphere  connecting 
the  two  points. 


Given  two  points,  A  and  5,  on  the  sphere,  and  AB  the  minor 
arc  of  the  great  circle  joining  them. 

To  prove  that  AB  is  the  shortest  curve  on  the  sphere  connect- 
ing A  and  B. 

Proof.  Select  any  point  C  on  the  arc  AB.  With  A  as  a  center 
and  with  AC  as  radius  construct  the  small  circle  S.  Similarly, 
with  B  as  a  center  and  EC  as  radius  construct  T.  §  639 

The  circles  S  and  T  have  only  the  point  C  in  common. 

For,  take  D,  any  point  on  S  other  than  C. 
In  the  spherical  AABD,  AD  +  DB  >  A  C  +  CB.        §  632 
But  AD=AC.  §638 

Therefore  DB  >  CB.  Why  ? 

Hence  D  is  not  on  T,  and  consequently  is  not  common  to  S  and  T. 

Now  let  AFB  be  any  curve  on  the  sphere  connecting  A  and  B 
which  does  not  contain  C. 

It  must  cut  S  and  T  in  distinct  points  K  and  L,  by  the  first 
part  of  the  proof. 

Consider  the  curve  which  might  be  drawn  connecting  A  and  C, 
which  is  congruent  to  A  A';  and  the  curve  similarly  connecting 
B  and  C,  which  is  congruent  to  LB. 

The  sum  of  the  lengths  of  these  curves  is  less  than  AKLB  by 
the  length  of  KL. 


BOOK  VIII  475 

Hence  there  is  a  curve  on  the  sphere  through  C  joining  A  and 
B,  which  is  shorter  than  AFB. 

But  since  C  was  any  point  on  the  minor  arc  AB,  the  shortest 
curve  on  the  sphere  connecting  A  and  B  must  contain  all  points 
of  AB,  and  hence  coincide  with  it. 

647.  Distances  on  a  sphere.    We  are  now  justified  in  call- 
ing the  minor  arc  of  the  great  circle  joining  two  points  on 
the  sphere  the  shortest  distance  between  them,  and  in  taking 
the  great    circle  as  the  figure  in  spherical  geometry  which 
corresponds  to  the  straight  line  in  plane  geometry. 

QUERY  1.  How  many  points  are  required  to  determine  a  circle  in  the 
plane  ?  Mention  any  exception. 

QUERY  2.  How  many  points  are  required  to  determine  a  small  circle 
on  the  sphere  ?  Mention  any  exception. 

QUERY  3.    What  use  is  made  in  navigation  of  the  fact  stated  in  §  646  ? 

648.  Assumption  of  free  motion  on  a  sphere.  Since  the  sphere 
has  the  same  curvature  throughout,  it  follows  that  figures  on 
the  sphere  may  be  moved  from  place  to  place  upon  it  with- 
out altering  their   size   or   shape.    This   corresponds   to   the 
important  assumption  of  plane  geometry  contained  in  §  20. 

649.  Restrictions  on  spherical  geometry.    In  two  important 
respects  geometry  on  the  sphere  differs  from  geometry  in  the 
plane. 

In  the  plane,  triangles  whose  parts  are  equal  each  to  each 
but  arranged  in  opposite  order,  like  those  in  the  adjacent 
figure,  can  be  brought  into 
coincidence  by  turning  one 
of  them  over  in  space  and 
applying  the  equal  parts  to 
each  other.    Symmetric  tri- 
angles  on  the  sphere  cannot 
be  brought  into  coincidence  in  this  way,  as  one  can  easily 
see  by  cutting  out  the  symmetric  triangles  from  the  peel 


476  SOLID  GEOMETKY 

of  an  orange.  Although  symmetric  spherical  triangles  are 
not  congruent,  they  are  equal  in  area  (§  644). 

Since  every  pair  of  great  circles  on  a  sphere  meet,  there  is 
nothing  on  the  sphere  corresponding  to  parallel  lines  in  the 
plane.  For  parallel  lines  never  meet  however  far  they  are 
produced.  Hence  theorems  in  plane  geometry  which  depend 
either  on  the  existence  of  parallel  lines  or  on  the  parallel 
assumption  (§  45)  cannot  be  carried  over  into  spherical  geom- 
try.  If,  however,  we  are  careful  to  avoid  such  theorems,  we 
may  state  a  large  number  of  theorems  from  plane  geometry 
which  are  true  on  the  sphere. 

The  following  theorems  of  spherical  geometrv  may  be  stated 
without  proof,  since  the  corresponding  theorems  in  plane  geom- 
etry do  not  depend  on  the  notion  of  parallels. 

1.  At  a  point  in  a  great  circle,  one  and  only  one  great  cir- 
cle can  be  drawn  perpendicular  to  it. 

2.  Vertical  angles  are  equal. 

3.  Two  right  spherical  triangles  are  congruent  if  the  hypot- 
enuse and  an  adjacant  angle  of  one  are  equal  respectively  to 
the  hypotenuse  and  an  adjacent  angle  of  the  other,  and  if 
the  corresponding  parts  are  arranged  in  the  same  order. 

4.  Two  spherical  triangles  on  the   same   sphere   are  con- 
gruent if  two  sides  and  the  included  angle  are  equal  respec- 
tively to  two  sides  and  the  included  angle  of  the  other  and  if 
the  corresponding  parts  are  arranged  in  the  same  order  in  the 
two  triangles. 

5.  Two  spherical  triangles  on  the  same  sphere  are  congru- 
ent if  a  side  and  the  adjacent  angles  of  one  are  equal  to  an 
angle  and  two  adjacent  sides  of  the  other,  and  if  the  corre- 
sponding parts  are  arranged  in  the  same  order. 

It  should  be  noted  that  4  and  5  may  be  proved  by  superposition 
in  precisely  the  same  manner  as  the  corresponding  theorems  in 
plane  geometry. 


BOOK  VIII  477 

EXERCISES 

71.  Two  spherical  triangles  on  the  same  sphere  are  symmetric 
if  two  sides  and  the  included  angle  in  one  are  equal  to  two  sides 
and  the  included  angle  in  the  other,  and  if  the  corresponding  parts 
are  arranged  in  opposite  orders. 

HINTS.  Denote  the  given  triangles  by  Tl  and  7T2.  The  triangle  sym- 
metric to  7\  has  its  parts  arranged  in  the  same  order  as  the  correspond- 
ing parts  of  Tz,  and  by  4  above  is  congruent  to  T2.  Hence  7\  and  7'2 
are  symmetric. 

72.  Two  spherical  triangles  on  the  same  sphere  are  symmetric 
if  a  side  and  two  adjacent  angles  of  one  are  equal  to  a  side  and 
two  adjacent  angles  of  the  other,  and  if  the  corresponding  parts 
are  arranged  in  opposite  orders. 

73.  Find  four  theorems  other  than  those  given  on  page  476 
which  correspond  to  theorems  in  plane  geometry  and  whose  truth 
can  be  inferred  without  proof. 

74.  Find  four  theorems  of  plane  geometry  which  correspond 
to  propositions  in  spherical  geometry  which  are  not  true. 

650.  Sum  of  angles  in  a  spherical  triangle.    Since  parallels 
do  not  exist  on  the  sphere,  there  is  no  such  figure  as  a  spher- 
ical parallelogram,  trapezoid,  or  square.   The  theorem  of  plane 
geometry  that  the  sum  of  the  angles  in  a  triangle  equals  two 
right  angles  necessarily  depends  upon  the  parallel  assumption. 
Since  this  assumption  does  not  hold  upon  the  sphere,  one 
would  not  expect  the  sum  of  the  three  angles  of  a  spherical 
triangle  to  equal  180  degrees.    We  now  proceed  to  prove  the 
theorems  which  will  lead  us  to  the  facts  in  the  case  of  spherical 
triangles. 

651.  Quadrant.    The  arc  of  a  great  circle  subtended  by  a 
right  angle  at  the  center  of  a  sphere  is  called  a  quadrant. 

652.  Corollary.    The  polar   distance   of  a  great   circle   is  a 
quadrant. 


478 


SOLID  GEOMETRY 
Theorem  14 


653.  If  two  points  are  taken  a  quadrant's  distance  from 
a  given  point,  they  determine  the  great  circle  of  which 
the  given  point  is  the  pole. 


Given  the  point  P  on  a  sphere,  and  two  other  points  of  the 
sphere,  A  and  B,  such  that  PA  and  PB  are  both  quadrants. 

To  prove  that  the  great  circle  of  which  AB  is  an  arc  has  P 
for  its  pole. 

Proof.    Pass  the  planes  of  the  great  circles  determined  by  PA, 
PB,  and  AB  respectively. 

These  planes  intersect  at  0,  the  center  of  the  sphere.        §  596 

Z.POA  =^POB=  90°.  Why  ? 

Therefore  PO  is  _L  to  the  plane  of  AB,  §  415 

and  P  is  the  pole  of  the  great  circle  of  which  AB  is  an  arc.       §  637 

654.  Corollary.    Construct  on  the  sphere  a  great  circle  determined 
by  two  given  points. 

HINTS.  Let  the  given  points  be  A  and  B.  Con- 
struct the  point  of  intersection  of  the  great  circles 
of  which  A  and  B  are  poles  (§  640),  and  apply  §  653. 

EXERCISE  75.    If  two  great  circles   on   the 
same  sphere  are  both  perpendicular  to  a  given 
great   circle  on  that  sphere,  they  meet  at  the  poles  of  the  given 
great  circle. 


BOOK  VIII  479 

Theorem  15 

655.  A  spherical  angle  is  measured  by  the  arc  of  the 
great  circle  of  which  its  vertex  is  a  pole,  and  which  is 
included  between  its  sides,  produced  if  necessary. 

P 


Given  any  spherical  angle  XPY,  and  A  and  B  the  points  where 
the  sides  of  this  angle,  produced  if  necessary,  meet  the  great 
circle  of  which  P  is  the  pole. 

To  prove  that       Z.XPY  is  measured  ly  arc  AB. 

Proof.  Draw  the  tangents  EP  and  SP  to  the  great  circles  AP 
and  EP  respectively. 

Then  Z.XPY  is  measured  by  ^.RPS.                   §  626 

But  RP  and  SP  are  both  _L  to  PO.                     §  192 

Also  A  O  and  BO  are  both  _L  to  PO.                    Why  ? 

Therefore  Z  A  OB  =  Z  RPS.                                Why  ? 

But  /.AGE  is  measured  by  the  arc  AB.               Why  ? 

Therefore  /.RPS  or  Z.XPY  is  measured  by  the  arc  AB. 

656.  Corollary.  If  one  great  circle  passes  through  a  pole  of 
another,  the  circles  are  perpendicular  to  each  other. 

QUERY.  If  the  angle  XPYin  the  figure  for  Theorem  15  is  40°,  how 
many  degrees  are  there  in  the  sum  of  the  angles  of  the  triangle  PA  B  ? 

EXERCISE  76.  If  one  vertex  of  a  spherical  triangle  is  the  pole 
of  the  opposite  side,  prove  that  the  sum  of  the  angles  of  the 
triangle  equals  the  sum  of  its  sides,  each  measured  in  degrees. 


480 


SOLID  GEOMETRY 


NOTE.  Since  the  measure  of  both  the  dihedral  angle  and  the  spher- 
ical angle  are  denned  in  terms  of  certain  plane  angles,  it  follows  that 
they  are  both  expressed  numerically  in  terms  of  the  units  which  meas- 
ure plane  angles,  namely,  degrees,  minutes,  and  seconds.  The  arc  of  a 
circle  is  also  measured  in  terms  of  these  units.  But  this  fact  does  not 
imply  that  these  magnitudes  are  of  the  same  kind,  any  more  than  the 
measure  of  the  height  of  houses,  trees,  and  mountains  in  terms  of  feet 
implies  any  similarity  in  their  geometric  form. 

QUERY  1.  What  is  the  locus  of  points  a  quadrant's  distance  from  a 
given  point  on  a  sphere? 

QUERY  2.  Is  each  angle  of  any  spherical  triangle  measured  by  the 
side  opposite  it  ? 

QUERY  3.  Can  a  spherical  triangle  be  constructed  so  that  each 
angle  has  the  same  measure  as  its  opposite  side? 

QUERY  4!  Can  a  spherical  triangle  be  constructed  so  that  two,  but 
not  three,  angles  are  measured  by  the  sides  opposite  them  ? 

657.  Polar  triangle.  Let  ABC  be  a  spherical  triangle.  Let 
A'  be  the  pole  of  the  great  circle  of  which  BC  is  an  arc, 
which  is  no  more  than  a  quadrant's  distance  from  A,  Let 
B'  and  C'  be  similarly  chosen  with  respect  to  the  other  sides 
of  ABC.  Then  A'B'C'  is  called  the  polar  triangle  of  ABC. 


An  inspection  of  the  above  figures  will  show  that  A'B'C' 
will  lie  entirely  outside,  or  entirely  inside,  ABC,  according  as 
the  sides  of  ABC  are  all  less  than  a  quadrant  or  all  greater 
than  a  quadrant.  If  at  least  one  side  of  ABC  is  less  than  a 
quadrant,  while  at  least  one  side  is  greater  than  a  quadrant, 
its  polar  triangle  will  overlap  it. 


BOOK  VIII  481 

Theorem  16 

658.  If  A'B'O'  is  the  polar  triangle  of  ABC,  then  ABC 
is  the  polar  triangle  of  A'B'C'. 


Given  any  triangle  ABC  and  its  polar  triangle  A'B'C'. 

To  prove  that    ABC  is  also  the  polar  A  of  A'B'C'. 

Proof.    C"  is  the  pole  of  AB,  and  B'  is  the  pole  of  AC.       §657 

Hence  A  is  a  quadrant's  distance  from  B'  and  from  C'.  §  652 

Hence  A  is  the  pole  of  B'C'.  §  653 

Since  A  is  in  the  same  hemisphere  with  A',  by  hypothesis,  it 
is  one  vertex  of  the  polar  A  of  A'B'C1.  §  657 

Similarly,  B  and  C  are  vertices  of  the  polar  A  of  A'B'C'. 

Therefore          ABC  is  the  polar  A  of  A'B'C'. 

QUERY  1.  Is  there  any  triangle  on  the  sphere  which  is  its  own  polar 
triangle  ? 

QUERY  2.  If  two  sides  of  a  triangle  are  quadrants,  does  it  bear  any 
particular  relation  to  its  polar  triangle  ? 

QUERY  3.  If  one  side  of  a  spherical  triangle  is  a  quadrant,  does  it 
bear  any  particular  relation  to  its  polar  triangle  ? 

QUERY  4.  If  two  angles  of  a  spherical  triangle  are  right  angles, 
does  the  triangle  bear  any  particular  relation  to  its  polar  triangle  ? 

EXERCISE  77.  Construct  the  polar  triangle  of  a  given  triangle, 
giving  a  reason  for  each  step. 


482  SOLID  GEOMETKY 

Theorem  17 

659.  Each  angle  of  a  spherical  triangle  is  the  supple- 
ment of  the  side  lying  opposite  it  in  its  polar  triangle. 


Given  any  spherical  triangle  ABC  and  its  polar  triangle  A'B'C1. 
To  prove  that 
Z.A  +  B'C'=180°,  ZB+A'Cr=180°,  Z C  +  B'A'  =  180°. 

Proof.  Let  F  and  //  be  the  intersections  of  B'C'  with  AB  and 
A  C  produced,  respectively. 

Now                       C'F  =  90°  and  HB'  =  90°.  §  657 

C'F  +  HB'  =  C'H  +  HF  +  HF  +  FB'  =  180°.  Why  ? 

But                                HF  measures  Z.A.  §655 

Hence  (C'H  +  HF  +  FB')  +  HF  =  C'B'  +  Z.A=  180°.  Why  ? 
Similarly,  Z.B  +A'C'  =  180°  and  Z C  +  .4 'B'  =  180°. 

QUERY  1.  If  a  spherical  triangle  is  equilateral,  what  can  be  said  of 
its  polar? 

QUERY  2.  If  a  spherical  triangle  is  equiangular,  what  can  be  said  of 
its  polar  ? 

QUERY  3.  If  a  spherical  triangle  is  isosceles,  what  can  be  said  of 
its  polar? 

QUERY  4.  If  a  spherical  triangle  has  two  angles  equal  to  each  other, 
what  can  be  said  of  its  polar  ? 

QUERY  5.  If  all  of  the  angles  of  a  spherical  triangle  are  right 
angles,  what  can  be  said  of  its  polar  ? 


BOOK  VIII  483 

Theorem  18 

660.  Two  spherical  triangles  on  the  same  sphere  which 
have  three  angles  of  one  equal  to  three  angles  of  the  other 
are  congruent  or  symmetric,  according  as  the  correspond- 
ing parts  are  arranged  in  the  same  or  in  opposite  orders. 

X 


Case  L    When  the  parts  are  arranged  in  the  same  order. 

Given  the  spherical  triangles  ABC  and  XYZ,  in  which  angle  A 
equals  angle  X,  angle  B  equals  angle  F,  angle  C  equals  angle  Z, 
and  the  parts  A,  B,  and  C  follow  each  other  in  the  same  order 
as  the  parts  JJT,  7,  and  Z. 

To  prove  that  A  ABC  =  AXYZ. 

Proof.    Construct  the  polar  triangles  A'B'C'  and  X'Y'Z'. 

A  'B'  =  180°  -  C,  X '  ¥'  =  180°  -  Z,  etc.  §  659 

Therefore   A'B'  =  X'Y',  B'C1  =  Y'Z',  and  C'A'  =  Z'X'.    Why? 

Hence  AA'B'C'  is  congruent  to  AX'Y'Z',  §  635 

and  Z.4 '  =  ZZ',  Z£'  =  Z  Y',  and  Z  C '  =  ZZr.  §  27 

Hence        EC  =  YZ,  CA  =  ZX,  and  AB  =  XY.  §  659 

Therefore  AABC  =  AXYZ.  §  635 

Case  II.    When  the  parts  are  arranged  in  opposite  orders. 
Denote  the  given  triangles  by  T^  and  T2. 

Any  triangle,  such  as  S,  which  is  symmetric  to  Tl  has  its  angles 
equal  to  those  of  T^  and  the  corresponding  parts  arranged  in 
the  same  order  as  those  in  jT2. 

Hence  the  triangle  S  is  congruent  to  Tf  Case  I. 

Therefore,  since  Tl  and  S  are  symmetric,  it  follows  that  Tl  and 
T^  are  symmetric. 


484  SOLID  GEOMETRY 

NOTE.  It  is  observed  that  this  theorem  is  in  striking  contrast  to 
the  corresponding  theorem  in  plane  geometry.  Since  congruence  (or 
symmetry)  follows  from  equality  of  angles  in  two  spherical  triangles, 
there  is  no  such  thing  as  similar  triangles  in  spherical  geometry.  Hence 
it  is  possible  to  construct  only  one  triangle  on  a  sphere  whose  angles 
are  known. 

EXERCISES 

78.  If  two  angles  of  a  spherical  triangle  are  equal,  the  triangle 
is  isosceles. 

79.  If  the  three  angles  of  a  spherical  triangle  are  equal,  the 
triangle  is  equilateral. 

80.  If  two  trihedral  angles  have  the  dihedrals  of  one  equal  to 
the  dihedrals  of  the  other,  the  corresponding  face  angles  are  equal 
and  the  trihedrals  are  either  symmetric  or  congruent. 

Theorem  19 

661.  The  sum  of  the  angles  of  a  spherical  triangle  is 
greater  than  180°  and  less  than  o40°. 


Given  the  spherical  triangle  ABC. 

To  prove  that     180°  <Z.A  +  ^B  +  ^C<  540°. 

Proof.  Construct  the  polar  AA'B'C'. 

Then  Z.A  =  180°  —  B'C'.  §  659 


BOOK  VIII 


485 


Adding,  £A  +  Z£  +  Z  C  -  540°  -  (.4  '5'  +  4'C"  +  B'C").. 
Since  .4  '5'  +  .4'C"  +  B'C'  cannot  be  zero  (or  negative), 


But 
Hence 


,4'jBf-f,4'C'-{-.B'C''<3600. 
180°<  Z.4  +  Z  JB  +  Z  C  <  540°. 


§631 
Why? 


QUERY  1.  Describe  the  appearance  of  a  spherical  triangle  the  sum 
of  whose  angles  is  181°. 

QUERY  2.  Describe  the  appearance  of  a  spherical  triangle  the  sum 
of  whose  angles  is  nearly  540°. 

QUERY  3.  Could  an  equiangular  triangle  each  of  whose  angles  is 
60°  exist  on  a  sphere? 

EXERCISE  81.  The  sum  of  the  angles  of  a  spherical  polygon  of 
n  sides  is  greater  than  2(n  —  2)  right  angles. 

662.  Birectangular  triangle.    A  spherical  triangle  two  of 
whose  angles  are  right  angles  is  called  a  Urectangular  triangle. 

The  arc  between  the  two  right  angles  is 
called  the  base,  the  opposite  vertex  is  called 
the  vertex,  of  the  triangle. 

QUERY  1.  Why  does  the  vertex  angle  of  a 
birectangular  triangle  contain  the  same  number 
of  degrees  as  the  base? 

QUERY  2.  Why  is  the  vertex  of  a  birectangular 
triangle  the  pole  of  the  base  ? 

663.  Lune.    A  lune  is  the  portion  of  a 
sphere   included  between  two  great  semi- 
circles which  meet  each  other. 

The  entire  sphere  may  be  considered  a  lune 
whose  angle  is  360°. 

664.  Spherical  degree.    The  birectangular 
triangle  whose  third  angle  is  1°  is  called  a 
spherical  degree. 


486  SOLID  GEOMETRY 

The  spherical  degree  on  a  given  sphere  is  taken  as  the  unit 
of  area  in  measuring  the  areas  of  figures  on  that  sphere. 

It  should  be  noted  that  the  number  of  square  inches  in  a  spher- 
ical degree  depends  on  the  radius  of  the  sphere.  Hence  a  given 
spherical  degree  is  not  a  unit  that  can  be  used  for  all  spheres. 

665.  Corollary.    A  lune  whose  angle  is  n°  contains  2n  spher- 
ical degrees. 

QUERY  1.    How  many  spherical  degrees  are  there  in  a  hemisphere  ? 

QUERY  2.    How  many  spherical  degrees  are  there  in  the  entire  sphere  ? 

QUERY  3.  How  many  spherical  degrees  are  there  in  a  lune  whose 
angle  is  (a)  1°,  (6)30°? 

QUERY  4.  Is  there  any  spherical  triangle  which  contains  three  right 
angles  ?  If  so,  how  many  spherical  degrees  does  it  contain  ? 

666.  Spherical  excess.    The  number  of  degrees  by  which 
the  sum  of  the  angles  of  a  spherical  triangle  exceeds  180°  is 
called  the  spherical  excess  of  the  triangle. 

QUERY  1.   What  is  the  spherical  excess  of  a  spherical  degree? 

QUERY  2.  What  is  the  spherical  excess  of  a  triangle  all  of  whose 
angles  are  right  angles? 

QUERY  3.  Why  must  the  spherical  excess  of  any  triangle  be  less 
than  360°? 

QUERY  4.  Why  must  the  spherical  excess  of  any  triangle  be  a  posi- 
tive number? 

EXERCISES 

82.  Construct  a  birectangular  triangle  containing  30  spherical 
degrees,  giving  a  reason  for  each  step. 

83.  Prove  that  two  spherical  degrees  on  two  different  spheres 
are  in  the  same  ratio  as  the  square  of  the  radii. 

84.  The  area  of  a  lune  is  to  the  area  of  the  sphere  as  the  angle 
of  the  lune  is  to  four  right  angles. 

85.  How  many  square  inches  are  there  in  a  spherical  degree 
on  a  sphere  whose  radius  is  6  inches  ? 


BOOK  VIII  487 

Theorem  20 

667.  The  area  of  a  spherical  triangle  is  equal  to  its 
spherical  excess  if  the  unit  of  area  is  the  spherical 
degree. 


Given  the  spherical  triangle  ABC  whose  spherical  excess  is 
denoted  by  E. 

To  prove  that  the  area  of  ABC  is  E  spherical  degrees. 

Proof.  Produce  the  sides  of  ABC  so  as  to  form  three  complete 
great  circles. 

Lune  A  CRB  =  2  A  spherical  degrees.  §  665 

Lune  BCSA  =  2  B  spherical  degrees. 

Lune  CBTA  =2  C  spherical  degrees.  Why? 

But  ACRS  =  ABAT.  §  644 

Now  lune  CBTA  =  A  ABC  +  ABA  T. 

Hence  A  ABC  +  ACSR  =  2  C  spherical  degrees. 

Also       A  ABC  +  ACAS  =  lune  BCSA  =  2  B  spherical  degrees, 
and  A  ABC  +A  CRB  =  lune  A  CRB  =  2  A  spherical  degrees. 

Adding, 

2  A  ABC  +  hemisphere  CABSR  =  2  (A  +  B  +  C)  spherical  degrees, 
or  2  ABC -f  360  spherical  degrees  =  2(A  +  B+C)  spherical  degrees. 

Hence     ABC  =  (A+B+C  - 180)  spherical  degrees.      Why ? 

But  4+5+C-180=:£.  §666 

Therefore  ABC  =  E  spherical  degrees. 


488  SOLID  GEOMETRY 

668.   Corollary.    The  area  of  a  spherical  triangle  expressed 
in  units  of  plane  area  is 


where  E  and  r  denote  the  spherical  excess  and  the  radius  of  the 
sphere  respectively. 

4  7T/"' 

HINT.    The  area  of  a  spherical  degree  on  a  sphere  of  radius  r  is  —  ;  —  • 

EXERCISES 

86.  The  area  of  a  spherical  triangle  is  to  the  area  of  the  sphere 
as  its  spherical  excess  is  to  8  right  angles. 

How  many  spherical  degrees  in  the  triangles  whose  angles  are 

87.  48°,  123°,  96°  ?  89.  90°,  110°,  167°  ? 

88.  156°,  197°,  43°  ?  90.  75°  each  ? 

If  the  radius  of  a  sphere  is  6  inches,  how  many  square  inches 
in  a  lune  whose  angle  is 

91.  334°?  93.  27°  45'? 

92.  96°?  94.  43°  37'  30"? 

If  the  radius  of  a  sphere  is  8  inches,  how  many  square  inches 
in  a  triangle  whose  angles  are 

95.  86°,  49°,  135°?  97.  29°,  150°,  74°? 

96.  128°,  137°,  196°  ?  98.  48°  17',  89°  15',  163°  ? 

99.  Find  the  angle  of  a  birectangular  triangle  which  is  equal 
in  area  to  a  zone  of  one  base  whose  altitude  is  one  half  the  radius 
of  the  sphere. 

100.  What  is  the  area  of  a  spherical  degree  on  the  earth? 
(r  =  4000  miles.) 

REVIEW  EXERCISES 

101.  Construct  a  line  tangent  to  a  sphere  from  an  exterior 
point,  giving  a  reason  for  each  step. 

102.  A  cubic  foot  of  ivory  weighs  114  pounds.   Find  the  weight 
of  a  billiard  ball  2^  inches  in  diameter. 


BOOK  VIII  489 

103.  Two  spheres  of  lead,  of  radii  2  and  3  inches  respectively, 
are  melted  into  a  cylinder  of  revolution  of  radius  1  inch.    Find 
the  altitude  of  the  cylinder. 

104.  The  surface  of  a  hemispherical  dome  whose  diameter  is 
36  feet  is  to  be  covered  with  gold  leaf  which  costs  15  cents  per 
square  inch.    What  must  be  paid  for  the  gold  leaf  ? 

105.  A  spherical  shell  2  inches  thick  has  an  outer  diameter  of 
12  inches.    Find  its  volume. 

106.  Find  the  volume  of  a  sphere  inscribed  in  a  cube  whose 
volume  is  216  cubic  inches. 

107.  Find  the  ratio  of  the  surface  of  a  sphere  to  that  of  its  cir- 
cumscribed right  circular  cylinder. 

108.  A  wooden  sphere  weighs  200  pounds.    Find  the  diameter 
of  a  sphere  of  the  same  material  which  weighs  50  pounds. 

109.  The  diameter  of  one  iron  sphere  is  twice  that  of  another. 
What  is  the  ratio  of  their  weights  ? 

110.  The  area  of  a  certain  spherical  triangle  is  60  spherical 
degrees.    Its  angles  are  in  the  ratio  1:2:3.    Find  the  angles  of 
the  spherical  triangle. 

111.  Prove  that  two  lunes  on  unequal  spheres,  but  with  equal 
angles,  are  to  each  other  as  the  squares  of  the  radii  of  the  spheres. 

112.  Given  a  sphere.    Find  the  ratio  of  the  volume  of  an  in- 
scribed cube  to  that  of  a  circumscribed  cube. 

113.  Prove  that  the  sphere  inscribed  in,  and  the  sphere  circum- 
scribed about,  a  given  regular  tetrahedron  have  the  same  center, 
which  is  the  point  where  the  medians  of  the  tetrahedrons  meet. 

HINT.    See  Exercise  133,  Book  VII. 

114.  If  a  square  of  side  a  lies  with  all  its  vertices  on  the 
surface  of  a  sphere  of  radius  ?*,  how  far  from  the  center  of  the 
sphere  is  the  plane  of  the  square  ? 

115.  Prove  that  all  of  the  planes  which  make  equal  sections  of 
a  given  sphere  are  tangent  to  a  sphere  concentric  with  the  given 
sphere. 


490  SOLID  GEOMETRY 

116.  In  order  to  double  the  capacity  of  a  spherical  balloon, 
by  what  per  cent  must  the  area  of  the  material  in  its  surface  be 
increased  ? 

117.  Find  the  ratio  of  (1)  the  surfaces,  (2)  the  volumes,  of  a 
sphere  and  its  circumscribed  cube. 

118.  Through  a  fixed  point  P  outside  a  sphere  a  variable  line 
is  drawn  meeting  the  sphere  in  the  variable  points  A  and  B. 
Prove  that  PA  x  PB  is  constant.    Find  the  value  of  this  constant 
in  terms  of  r,  the  radius  of  the  sphere,  and  d,  the  distance  from 
P  to  the  center  of  the  sphere. 

119.  Four  spheres  6  inches  in  diameter  are  placed  in  a  square  box 
whose  inside  dimensions  are  12  inches.    In  the  space  between  the 
first  four  spheres  a  fifth  of  the  same  diameter  is  placed.  How  deep 
must  the  box  be  so  that  the  top  will  just  touch  the  fifth  sphere  ? 

120.  A  ball  18  inches  in  diameter  is  placed  in  the  corner  of 
a  room  where  the  walls  and  the  floor  are  at  right  angles.    Find 
the  diameter  of  another  ball  which  will  just  fit  back  of  the  first 
one,  touching  the  large  ball,  the  walls,  and  the  floor. 

121.  Using  a  method  analogous  to  that  of  plane  geometry, 
construct  a  spherical  triangle  each  of  whose  sides  is  60°. 

122.  Show  that  the  attempt  to  construct  a  spherical  triangle 
each  of  whose  sides  is  120°  does  not  lead  to  a  triangle. 

123.  Show  that  there  is  no  plane  section  of  a  sphere  and  its 
inscribed  cube  which  affords  a  great  circle  and  an  inscribed  square. 

124.  Show  that  there  are  three  plane  sections  of  a  sphere  and 
an  inscribed  octahedron  which  afford  a  circle  and  an  inscribed 
square. 

125.  Construct  a  sphere  of  given  radius  passing  through  three 
given  points. 

126.  Prove  that  the  six  planes  which  bisect  perpendicularly 
the  six  edges  of  a  tetrahedron  meet  in  a  point. 

127.  Construct  the  sphere  circumscribed  about  a  given  tetra- 
hedron. 


BOOK  VIII  491 

128.  The  diameter  of  the  earth  is  7960  miles.    That  of  the 
sun  is  860,000  miles.    (1)  Find  the  length  of  the  earth's  shadow. 
The  distance  of  the  moon  is  240,000  miles.    (2)  Find  the  diameter 
of  the  earth's  shadow  at  that  distance  from  the  earth. 

129.  If  the  moon  moves  3  miles  per  minute  with  respect  to 
the  earth  and  its  diameter  is  2160  miles,  what  is  the  longest 
time  that  an  eclipse  of  the  moon  can  last? 

130.  Through  a  given  fixed  line  construct  a  plane  cutting  a 
given  sphere  in  (1)  a  great  circle,  (2)  a  small  circle  of  prescribed 
radius,  (3)  a  point. 

131.  Prove  that  the  area  of  the  zone  of  a  sphere  of  radius  r, 
which  is  illuminated  by  a  point  of  light  a  distance  a  from  the 
surface  of  the  sphere,  is  2  7ra?^/(a  +  r). 

132.  The  diameter  of  the  earth  is  7960  miles,  and  that  of  the 
moon  is  2160  miles.    Compare  their  volumes. 

133.  Find  the  area  of  the  torrid  zone  if  its  width  is  47°. 
HINT.    Sin  231° -.398. 

134.  What  is  the  distance  of  the  horizon  on  a  calm  sea  from  a 
point  h  feet  in  height,  assuming  that  the  line  of  vision  is  in 
a  straight  line  ? 

135.  The  peak  of  Teneriffe  is  near  latitude  30°K    The  sun 
rising  in  the  exact  east  shines  on  its  summit  9  minutes  before  it 
shines  on  its  base.    How  high  is  the  mountain  ?   (Compare  the 
approximation  found  by  this  method  with  the  exact  height  given 
in  an  atlas.) 

136.  Prove  that  in  a  spherical  hexagon  the  sum  of  the  interior 
angles  is  >  4  and  <  8  right  angles. 

137.  The  sides  of  a  spherical  triangle  ABC  are  60°,  100°,  and  80° 
respectively.    How  many  degrees  in  the  vertex  angle  of  a  birec- 
tangular  triangle  of  the  same  area  on  the  same  sphere  ? 

138.  If  the  diameter  of  the  earth  is  taken  as  8000  miles,  and 
the  distance  of  the  sun  as  93,000,000  miles,  what  per  cent  of  the 
total  light  and  heat  of  the  sun  is  received  by  the  earth  ? 


INDEX 


Angle,  between  line  and  plane,  352  ; 
dihedral,  339;  plane,  339;  poly- 
hedral, 419 ;  trihedral,  420 

Bisector  of  dihedral  angle,  345 

Cavalieri's  theory,  369 

Circumscribed  prism,  385 

Circumscribed  pyramid,  418 

Circumscribed  sphere,  449 

Cone,  405;  axis  of,  408;  circular, 
408  ;  lateral  area  of,  412  ;  of  revo- 
lution, 416  ;  right  circular,  408  ; 
slant  height  of,  408 ;  spherical, 
460  ;  volume  of,  410  ;  volume  of 
spherical,  461 

Cones,  similar,  417 

Conical  surface,  405 

Construction,  operations  of,  314  ;  on 
the  sphere,  470 

Coplanar,  308 

Cube,  362 

Cylinder,  375  ;  axis  of,  381  ;  circular, 
375  ;  of  revolution,  381  ;  volume 
of,  378 

Cylinders,  similar,  381 

Cylindrical  surface,  375 

Diagonal,  363 
Dihedral  angle,  339 
Distance,  to  plane,  337  ;  on  a  sphere, 
475 


Element,  of  cone,  405 ;  of  cylinder, 

375 
Ellipse,  407 

Foot  of  a  line,  324 

Given,  meaning  of,  306  (footnote) 
Great  circle,  446 

Hemisphere,  448 
Hyperbola,  407 

Inscribed  prism,  378 
Inscribed  pyramid,  409 
Inscribed  sphere,  449 
Intersection,  309 

Line,  305 
Line-segment,  305 
Lime,  485 

Pappus,  Theorem  of,  383 
Parabola,  407 
Parallel  lines,  313 
Parallel  planes,  313 
Parallelepiped,  362  ;  right,  362 
Perpendicular,  325 
Perpendicular  planes,  341 
Perspective,  308 
Plane,  305 
Plane  angle,  339 
Polar  distance,  471 


493 


494 


SOLID  GEOMETRY 


Polar  triangle,  480 

Poll-,  469 

Polyhedral  angle,  419 

Polyhedron,  358  ;  convex,  358  ;  diag- 
onal of,  363  ;  regular,  424 

Prism,  358;  oblique,  360;  regular, 
361 ;  right,  360 ;  right  section  of, 
360 ;  truncated,  361 

Prismatoid,  435 

Projection,  of  area,  352;  of  line, 
349  ;  of  point,  349 

Proposition,  converse,  334 ;  direct, 
334 ;  opposite,  334 

Pyramid,  387 ;  altitude  of,  387 ; 
frustum  of,  395;  lateral  area  of, 
387;  regular,  399;  slant  height 
of,  399 

Pyramidal  surface,  388 

Quadrant,  477 

Rectangular  solid,  362 
Regular  polyhedron,  424 
Regular  prism,  361 
Regular  pyramid,  399 
Right  dihedral,  341 

Similar  cones,  417 
Similar  cylinders,  381 
Similar  figures,  435 


Similar  polyhedrons,  432 

Skew  lines,  354 

Skew  quadrilateral, '354 

Slant  height,  of  cone,  408 ;  of  frustum, 

401  ;  of  pyramid,  399 
Small  circle,  447 
Sphere,  443  ;  radius  of,  443  ;  tangent 

plane,  445  ;  volume  of,  443 
Spherical  angle,  465 
Spherical  cone,  460 
Spherical  degree,  485 
Spherical  excess,  486 
Spherical  geometry,  464  ;  restrictions 

on,  475 

Spherical  polygon,  464 
Spherical  sector,  460 
Spherical  segment,  460 
Symmetric  spherical  triangles,  466 
Symmetric  trihedrals,  431 

Tetrahedron,  397  ;  regular,  399 
Triangle,  bi-rectangular,  485  ;  polar, 

480 

Trihedral  angle,  420 
Truncated,  361 

Undefined  terms,  305 

Volume,  of  cone,  410;  of  cylinder, 
378  ;  of  rectangular  solid,  365 


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